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공학박사 학위논문

Study on improvements of near-eye

holography: form factor, field of view,

and speckle noise reduction

근안 홀로그래픽 디스플레이의 개선에 관한

연구: 폼 팩터, 시야 각, 스페클 노이즈

2018년 2월

서울대학교 대학원

전기·컴퓨터공학부

리 강

i

Abstract

Study on improvements of near-eye

holography: form factor, field of view,

and speckle noise reduction

Gang Li

Department of Electrical Engineering and Computer Science

College of Engineering

Seoul National University

This dissertation presents solutions targeting on the three fundamental issues

on near-eye holography: 1. Form factor, 2. Field of view (FOV), and 3.

Speckle noise.

For the form factor reduction of the holographic display, a

multifunctional holographic optical element (HOE) is developed to replace

the beam splitter and the eyepiece lens of the conventional holographic

display. As a result, it enables us to build a compact see-through holographic

display for augmented reality (AR). The feasibility of using HOE in the

holographic display is verified by analyzing some optical characteristics of the

HOE, based on the Kogelnik’s coupled-wave theory. In addition, a

compensation method is introduced to compensate the wavefront aberration

caused by use of the HOE.

To build a wide FOV holographic head mounted display, I proposed two

methods to increase the space bandwidth product (SBP) which can widen the

ii

FOV without losing eyebox size. In each optical path of +1st, 0th, -1st

diffraction order of a spatial light modulator (SLM), I place a shutter

synchronized with the SLM to display different holograms by temporally

dividing the frames of the SLM. The second proposed method is called

polarization selective holography which is a brand new hologram calculation

and encoding method. In this work, an SLM can modulate waves in different

modes through different states of wave retardation and polarization optical

elements. Two sets of wave plates and polarizers with optimized states are

placed in the path of +1st and -1st different diffraction orders of an SLM, so

that the SLM reconstructs two different holograms via each path. As a result,

the system achieves a two-fold increase in terms of SBP for a single SLM.

Furthermore, a curved hologram can provide wider FOV comparing to a

planar one, when they have the same SBP. A curved surface hologram can

provide wider FOV. Thus, curved holographic display has a great potential to

the near-eye holography. Since the curved surface SLM has not been

developed yet, as an example, I also proposed a numerical method of

spherical hologram computation of real objects in this chapter.

For the speckle noise issue, some analyses are carried out to compare the

quality of holographic reconstruction images using different light source such

as a light-emitting diode (LED) and a laser diode (LD). Both of them have

pros and cons. The main advantage of use of the LED in the holographic

display is it can reconstruct the holographic image with significantly less

speckle noise compared to laser or LD. However, an inherent problem caused

by use of the LED is that it is difficult to reconstruct images located far away

from the SLM. This problem is mitigated using a magnifier optical system. As

iii

a result, I demonstrate an LED based near-eye holographic display system not

only to reduce the speckle noise but also to provide sufficient expressible

depth range.

Keywords: Holographic display, augmented reality, virtual reality, form

factor, holographic optical element, field of view, space bandwidth product,

high-order diffraction, wavefront aberration, speckle noise, coherency

Student Number: 2013-31297

iv

Contents

Abstract ...................................................................................................... i

Contents .................................................................................................... iv

List of Tables ............................................................................................... vii

List of Figures ............................................................................................... viii

Chapter 1 Introduction ................................................................................... 1

1.1 Overview of near-eye displays............................................................ 1

1.2 Motivation of this dissertation ............................................................ 7

1.3 Scope and organization ....................................................................... 9

Chapter 2 Form factor reduction of near-eye holography using a

multifunctional holographic optical element ............................ 11

2.1 Introduction of form factor issues in near-eye holographic display .. 11

2.2 Introduction of volume hologram ...................................................... 13

2.2.1 Diffraction efficiency: Coupled-wave theory......................... 14

2.2.2 Condition to be met for using holographic optical element in

holographic display ................................................................. 19

2.3 Multifunctional holographic optcal element fabrication and

application ......................................................................................... 27

2.3.1 Design of mirror-lens holographic optical element ................ 27

2.3.2 Fabrication of mirror-lens holographic optical element ......... 30

2.3.3 System implementation and experimental results .................. 33

2.4 Wavefront aberration compensation method for holographic optical

element .............................................................................................. 37

2.5 Summary and discussion .................................................................. 44

Chapter 3 Field of view expansion of near-eye holographic display ....... 45

3.1 Overview of field of view expansion for near-eyeholography ......... 45

3.1.1 Investigation on constrains in expanding field of view for

near-eye holography ................................................................ 45

v

3.1.2 Previous space bandwidth product enhancement methods for

holographic display ................................................................. 50

3.2 Temporal multiplexing of high-order diffraction guided by

holographic attenuating mirror ......................................................... 53

3.2.1 Design of holographic attenuating mirror .............................. 54

3.2.2 System imprelentation and expperiment results .................. 60

3.3 Polarization selective holography ..................................................... 67

3.3.1 Principle of polarization selective holography ....................... 67

3.3.2 Amplitude combination map optimization using genetic

algorithm ................................................................................. 71

3.3.3 Simulation and experimental results ...................................... 73

3.4 Synthesis of computer generated spherical hologram....................... 76

3.4.1 Spatial spectral bandwidth of spherical hologram ................. 76

3.4.2 Spherical hologram calculation of real objects ...................... 81

3.4.3 Simulation and experimental results ...................................... 88

3.5 Summary and discussion .................................................................. 95

Chapter 4 Light source selection for near-eye holography ........................ 97

4.1 Coherence of light source ................................................................. 97

4.1.1 Temporal coherence ............................................................... 97

4.1.2 Spatial coherence ................................................................... 98

4.2 Image sharpness for holographic display ........................................ 101

4.3 Ideal optical design for LED based near-eye holography ............... 105

4.4 Summary and discussion ................................................................ 108

Chapter 5 Conclusion .................................................................................. 109

Bibliography ................................................................................................. 111

Appendix ..................................................................................................... 121

vi

List of Tables

Table 1.1 Features of expressible depth range for 3D displays. ........................ 6

Table 2.1 Zernike polynomial coefficients of compensation. ......................... 42

Table 3.1 Definition of the FOV and eyebox size in the two types of near-eye

holography. .................................................................................... 47

Table 3.2 Parameters of the relay lenses and virture SLMs. ........................... 65

Table 4.1 Features of the two light sources (same spatial coherence) for

holographic display. .................................................................... 105

vii

List of Figures

Figure 1.1 Features of Human visual perception in accordance with FOV ...... 3

Figure 1.2 The objective and the scope of the dissertation ............................. 10

Figure 2.1 (a) Optical configuration of conventional holographci display. (b)

Concept of the multifunctional HOE............................................. 11

Figure 2.2 Diffraction in a volume grating. ..................................................... 13

Figure 2.3 Model of diffraction in a volume grating ....................................... 15

Figure 2.4 Diffraction efficiency distribution (Bragg-matched condition)

according to the different angles of reference and signal waves. (a)

Transmission grating and (b) reflection grating. ........................... 20

Figure 2.5 Angular selectivity of (a) reflection grating and (b) transmission

grating. .......................................................................................... 22

Figure 2.6 Experiment setup for validate the feasibility of HOE for

hologrpahic display ....................................................................... 23

Figure 2.7 Experiment and simulation results for validate the feasiblity of

HOE for holographic display. (a) Angular deviation of

reconstructed wave and (b) diffraction efficiency according to the

angular deviation of probe wave. .................................................. 24

Figure 2.8 Resloving power comparison between a normal lens and a HOE

lens. ............................................................................................... 25

Figure 2.9 Aberration comparison between a normal lens and a HOE lens. ... 26

Figure 2.10 Schematic diagram of the MLHOE recording and reconstruction

procedures. .................................................................................... 28

Figure 2.11 Experiment to investigate aberrations of HOE lens the hologram

reconstruction procedure. .............................................................. 29

Figure 2.12 Diffraction efficiency distribution according to exposure energies.

....................................................................................................... 31

viii

Figure 2.13 Experiment setup for (a) 1st step and (b) 2nd step of the MLHOE

recording process........................................................................... 33

Figure 2.14 Protopype of the compact holographic display using MLHOE. .. 34

Figure 2.15 Experimental results: camera focus at (a) the Fourier plane of the

SLM, (b) the MLHOE, and (c) the real object “Rubik’s cube,” (d)

Camera is set to have a large depth of focus. ................................ 35

Figure 2.16 Experimental setup. ..................................................................... 39

Figure 2.17 (a) Extracted phase of LHOE (b) Zernike polynomials (c) Zernike

polynomials of aberration term. .................................................... 41

Figure 2.18 Zernike polynomials of correction wavefront (b) phase of

wavefront (c) converted gray-scale image. ................................... 41

Figure 2.19 PSF images (a) before compensation on Bragg matching (b) after

compensation on Bragg matching (c) before compensation on

Bragg mismatching (d) after compensation on Bragg mismatching

. ...................................................................................................... 42

Figure 3.1 Optical designs for near-eye holography. ...................................... 46

Figure 3.2 Experiment schematic diagram and setup of a wide FOV near-eye

holographic display. ...................................................................... 47

Figure 3.3 Experimental results: (a) reconstructed image when the camera

focuses at 5 diopter. (b) Reconstructed image when the camera

focuses at 0 diopter. ....................................................................... 48

Figure 3.4 FOV and DOF variation according to eyebox size in a holographic

display (SLM pixel number 1920×1080, pixel pitch: 8 um) . ....... 49

Figure 3.5 Schematic diagram of proposed method. ....................................... 53

Figure 3.6 Grating detuning effect caused by dimensional shrinkage. ........... 54

Figure 3.7 Vector space diagram for original grating and detuned grating cases.

....................................................................................................... 55

Figure 3.8 Schematic of the (a) recording and (b) reconstruction processes .. 57

Figure 3.9 Diffraction efficiency according to exposure energy. .................... 60

ix

Figure 3.10 Guided three diffraction orders by the HOEs. ............................. 61

Figure 3.11 Measurement of FZP aberration. (a) Configuration and (b)

experimental setup. (c) Captured reconstruction images .............. 62

Figure 3.12 Comparisons of the reconstructed holograms: (a) without using

the compensation method and (b) using the compensation method.

....................................................................................................... 63

Figure 3.13 Configuration of proposed holographic display system. ............. 64

Figure 3.14 Experiment setup of the proposed holographic display system. .. 65

Figure 3.15 Experimental results of the reconstructed holograms by the

holographic display with increased SBP. ...................................... 66

Figure 3.16 Concept of the polarization selective holography. ....................... 67

Figure 3.17 Ideal case of ACM for an 8 bit gray level SLM. .......................... 69

Figure 3.18 Single-sideband encoding method for amplitude-only SLM. ...... 70

Figure 3.19 Flow chart of the genetic algorithm. ............................................ 72

Figure 3.20 ACM and HCM distribution at A1-A2 plane. ................................ 73

Figure 3.21 Simulation results using polarization selective holography. ........ 74

Figure 3.22 (a) Experiment setup. Reconstruction images: (b) -1st diffraction

mode, (c) +1st diffraction mode. .................................................... 75

Figure 3.23 (a) Position of the object point and the hologram surface. (b)

Amplitude and (c) phase distribution of the spherical hologram. . 77

Figure 3.24 Object points with the same distance from coordinate origin. ..... 79

Figure 3.25 Spherical and planar hologram for optical field of the same

FOV….. ......................................................................................... 80

Figure 3.26 Required number of samples for the spherical and the planar

hologram. d′= 0.18 m, r=1 m ........................................................ 81

Figure 3.27 Extracted depth maps of the object in multiple directions. (a) Colr

images of the object, (b) corresponding depth map images. ......... 82

Figure 3.28 Rotation of the initial point cloud sets. (a) Initial point clouds and

(b) rotated ones .............................................................................. 83

x

Figure 3.29 Acquired 3D model of the real object. ......................................... 85

Figure 3.30 Speherical flipping of the 3D point cloud by Eq. (3.24). The

object point cloud ˆTP and the flipped point cloud ˆ

fP . ................ 86

Figure 3.31 (a) Entire points without HPR and (b) the visible point set

processed by HPR at the hologram point Hr(90°, 90°).. ............... 87

Figure 3.32 Visible point set for each hologram point.. .................................. 87

Figure 3.33 Spherical hologram generated by one point object. (a) Amplitude

and (b) phase images. .................................................................... 89

Figure 3.34 Amplitude of the generated spherical hologram. ......................... 90

Figure 3.35 Reconstruction process of the selected region. ............................ 91

Figure 3.36 Reconstruction of the partial regions of the spherical hologram.

(a) Amplitude, (b) phase and (c) reconstruction image of the

selected hologram region. ............................................................. 93

Figure 3.37 (a) Reconstruction image from a partial region of the proposed

spherical hologram, (b) reconstruction image from the spherical

hologram generated without using the HPR algorithm.. ............... 94

Figure 4.1 Correlation between two light points from a light source. ............. 99

Figure 4.2 Comparison of image sharpness using light sources with different

coherencies. (a) How temporal coherencce and (b) spatial

coherence affect qualty of reconstruction images. ...................... 102

Figure 4.3 (a) Holograms reconstructed at different depths from an SLM using

LD and LED light source. (b) Spectrum of the light source. ...... 104

Figure 4.4 Optical design for the near-eye holography using a light source

with low temporal coherence.. .................................................... 106

Figure 4.5 Reconstructed hologram at different depth plane using the

proposed optical configuraation. ................................................. 107

1

Chapter 1 Introduction

1.1 Overview of near-eye displays

Since virtual reality (VR) and augmented reality (AR) have widely gained

spotlights, the near-eye display has recently become an attractive research

field. The near-eye displays have great potentials in various fields such as

education, teleconferencing, scientific visualization, entertainment, military,

and training [1]. So far, plenty of studies have been introduced in the both

academia and industry area. Recently, the launch of several commercialized

near-eye displays such as Oculus Rift, HTC VIVE, PlayStation VR,

Samsung Gear VR, Google Glass, and HoloLens, open the new era where

people can cross the real and the virtual world. However, those

commercialized products still have many problems. Some main issues for

realizing an ideal VR/AR device are as follows:

Form factor: One main reason that causes large and heavyweight form

factor is the embedded various optical elements. A head-mounted display

(HMD) requires magnifier optics and beam guiding optics. To provide a

sufficient field of view (FOV) the diameter of an eyepiece lens should be

large enough. So far, researchers have proposed many alternative ways to

replace the conventional heavyweight and large size imaging optics in the

HMD system. Currently, a popular schematic for getting a good form factor

of the AR/VR devices is the waveguide (or lightguide) combined with a set

of coupling gratings (in-coupling & out-coupling) [3-7]. The in-coupling

2

grating could be diffractive optical element (DOE) or holographic optical

element (HOE). The coupled light propagates through the waveguide (or

lightguide) via total internal reflection and it is out-coupled when it hits the

out-coupling grating. As a result, the use of DOE and HOE enables us to

reduce the form factor.

FOV: To provide an immersive experience, the HMD should provide a

sufficient large FOV. Figure 1.1 describes the features of human visual

perception in accordance with the FOV. As shown in Fig. 1.1, different FOV

range is used to perceive different properties of image. The foveal vision

zone, which has 3°-5° FOV, is the part of the retina that permits maximum

visual acuity [2]. In general, the eye rotation left and right from one’s normal

line of sight is 15 degrees, thus the total binocular FOV to perceive

immersive experience in VR is around 200°-220°. While an AR device does

not need such a large FOV, because the real scene already occupies most of

the observers’ FOV. The virtual image is just need to be displayed within a

specified region within the real scene.

3

Figure 1.1 Features of human visual perception in accordance with FOV

Eyebox: The eyebox determines how much the eyeball can move

up/down/left/right from the optimal position without significant degradation

in the image quality. In a HMD system, the image quality at the optimal

position is always the best, but if the eyebox is too narrow, the user will not

obtain a good image without tedious adjustments [11]. In addition, the

eyebox size should be decided in accordance with the FOV. For example, 10°

eye rotation requires about 2 mm eye pupil shifting in the corresponding

direction.

Aberration: Lenses and curved mirrors introduce a variety of optical

aberrations [8-10]. Chromatic aberrations occur due to refractive feature of

the lenses. Reflective optical elements such as concave mirrors do not induce

chromatic aberrations. Considering that full-color displays only have RGB

4

components, chromatic aberrations can be compensated by separately

predistorting R, G, and B planes at the expense of increased rendering costs.

This technique greatly contributes to flexibility in optical designs, resulting,

for example, in inexpensive wide FOV such as Oculus Rift. Spherical

aberrations are induced by the spherical shape of the lens surface. With

lateral shift of the eye, the image gets distorted and blurred. Similarly, field

curvatures caused blurred imagery in the periphery. Rendering based

predistorting technics are not effective to correct these aberrations. Instead,

aspheric and/or achromatic lenses can be used.

Focus cues: Depth perception occurs with monocular and/or binocular

depth cues. These cues can be further categorized into physiological and

psychological cues [12-15]. Physiological monocular depth cues include

accommodation, monocular convergence, and motion parallax.

Psychological monocular depth cues include apparent size, linear perspective,

aerial perspective, texture gradient, occlusion, shades, and shadows.

Binocular convergence and stereopsis are typical physiological and

psychological binocular depth cues, respectively. Binocular convergence is

related to the angle between two lines from a focused object to the both eyes,

while stereopsis is about the lateral disparity between left and right images.

Although current generation VR displays provide some of the depth cues,

however, focus cue is usually not supported. The decoupled focus cues and

vergence, which is called vergence accommodation conflict (VAC) induces

user uncomfortable physiological functions such as fatigue, double vision,

and eyestrain. [12-17]. To display natural three-dimension (3D) images,

researchers in academy and industry attempted to apply the existing 3D

5

display technics to the HMD systems for AR/VR. Some examples are as

follows:

Recently, some researchers applied retinal projection technic:

Maxwellian view display, to achieve a significant large depth of field (DOF)

for binocular disparity based displays. The method has all-in focus effect

because of the tiny exit pupil. As a result, this type display achieves the long

DOF by losing the resolution of the displayed images. Furthermore, because

of the all-in focus effect, in order to provide the focus cue, it requires

artificial blur for the rendered images based on the user’s convergence

information.

Varifocal display is typically categorized into two types i.e. varying lens

focal length [18, 19] and varying display position [20]. The former uses a

varifocal lens such as liquid lens to focus the display at different focal

distances depending on the user’s fixation point. The latter dynamically

translates the micro-display, which is mounted on a micro-controlled stage.

Both of them require eye-tracking system that is capable of tracking the

convergence of both eyes. In these methods, the latency between the

electronic devices is the main issue to provide a natural varifocal images.

Light field display describes the radiance of emitted light rays as a

function of both position and direction. The light field display can be

categorized into three types: parallax barrier based multiview display [21-23],

lens array based integral imaging display [24-28] and multilayer display [29-

32]. They provide multiple directional view images by scarifying the spatial

resolution of the display. Motion parallax is the dominant depth cue in the

light field displays while the most important factor to alleviate the VAC of

6

the HMD is focus cue. To date, many studies have reported about light field

based near-eye displays, however, there is no enough evidences to prove that

the technic indeed provides 3D images in any desired depth plane.

All the 3D display techniques mentioned about have a common problem

of providing sufficient focus cues to users. Compared with the other 3D

display technics, holography is regarded as the most natural way to provide

correct depth cues since it can reconstruct 3D images with full wavefront

information [35-49]. Holography does not have any tradeoff and constraint in

providing focus cues. The features of expressible depth range for the 3D

display technics mentioned above are listed in Table 1.

Table 1.1Features of expressible depth range for 3D displays

Trade-off (MTF vs. DOF) Constraint

Maxwellian view Yes Yes (Exit pupil size)

Varifocal No Yes (Frame rate)

Light field Yes No

Holography No No

7

1.2 Motivation of this dissertation

To be a practical method for HMD, holography confronts several

fundamental problems: 1. Form factor, 2. FOV, 3, Speckle noise. There are

few studies about these issues. This dissertation introduces several methods

that have been proposed targeting on these issues.

This dissertation introduces novel approaches using a multifunctional

HOE to replace a beam splitter and an eyepiece lens in the conventional

holographic display. In this study, I verified the feasibility of using HOE in

the holographic display through some analyses based on Kogelnik’s coupled-

wave theory. The method can be contributed to building a compact see-

through near-eye holographic display for AR [46, 47].

Furthermore, to build a wide FOV holographic HMD, I proposed two

methods to increase the SBP of the display. The methods can effectively

solve the trade-off between FOV and eyebox size in the conventional

holographic display. Especially, the second one called polarization selective

holography is a brand new method for hologram computation and encoding.

It enables us to modulate two different holograms using different diffraction

orders of a spatial light modulator (SLM) without any mechanical

movements. Furthermore, a spherical hologram method of real objects is

proposed. The method can be the basic computation architecture when

curved SLMs are available in the future.

Last, for speckle noise issue, I introduce some analyses on the light

source selection. To suppress the speckle noise, I measured the feasibility of

using LED light source for a near-eye holographic display. An inherent

8

problem caused by use of the LED can be mitigated by using a magnifier

optical system. This concept can be used as the basic model for the near-eye

holographic display.

The ultimate goal of the dissertation is to propose some effective

methods for developing more advanced near-eye holographic display.

9

1.3 Scope and organization

Chapter 1 gives an over view for the some important issues in near-eye

displays and explains why the holography should be the ultimate technic in

terms of focus cue ability.

In Chapter 2, a form factor reduction method is introduced using a

multifunctional HOE. Before explaining the proposed idea, some optical

characteristics of the HOE are measured to validate the condition to be met

for using HOE in the holographic display. Then, I introduce the design and

fabrication of the multifunctional HOE. A mirror and a lens are sequentially

recorded on a single hologram recording material. To avoid the crosstalk

between the two optical elements, the angles of the reference beam for the

two signal beams are separated to a sufficient large angle. Last, a wavefront

aberration method is proposed to precisely compensate the aberration caused

by use of the HOE.

In Chapter 3, two SBP enhancement methods for holographic display

are demonstrated. The first one mechanically divide an SLM’s frame into

three parts so that users can observe different reconstructed images through

+1st, 0th, -1st diffraction order of an SLM. The second method is named

polarization selective holography. The principle, simulation and system

implementation are introduced. Furthermore, a spherical hologram

computation method considering occlusion effect is introduced in this

chapter.

In chapter 4, different light source analyses on the reconstruction image

quality is demonstrated. The LED is selected as the alternative to suppress

10

the speckle noise. Then, an optical design that mitigates the inherent problem

for use of LED is demonstrated.

Finally, concluding remarks to this dissertation are presented in Chapter

5. The objective and the scope of this dissertation is depicted in Fig. 1.2.

Figure 1.2 The objective and the scope of the dissertation

11

Chapter 2 Form factor reduction of

near-eye holography using a

multifunctional holographic optical

element

2.1 Introduction of form factor issues in near-eye

holographic display

A conventional holographic display consists of a liquid crystal on silicon

(LCOS) SLM, a beam splitter and an imaging lens as shown in Fig. 2.1.

Figure 2.1 (a) Optical configuration of conventional holographic display. (b)

Concept of the multifunctional HOE.

One of the important issues to realize a see-through near-eye

holographic display is that the eyepiece part should be transparent and

compact enough. As shown in Fig. 2.1(a), the configuration is unable to

12

provide a see-through effect. Thus, I propose a multifunctional HOE named

with mirror lens HOE (MLHOE), which performs the functions of both beam

splitter and lens at the same time. The properties of the HOE recording

material enable us to use a transparent, lightweight, and thin optical element

as the eyepiece lens.

This chapter introduces the see-through holographic display system

realized by the MLHOE. First, I introduce the characteristics of a volume

hologram, and analyze the condition to be met for using HOE for

holographic display depending on the coupled-wave theory [51]. The

fabrication of the MLHOE is also included in this chapter. Finally, the

benchtop of the proposed system is demonstrated.

13

2.2 Introduction of volume hologram

Gratings behave differently depending on the relation between the period of

the finest fringe and the thickness of the recording medium. According to the

relation, it is generally categorized to thick or thin hologram.

If we consider a hologram consisting of a single sinusoidal grating with

grating plane normal to the surface of the emulsion, it behaves as a thick or

thin grating depending on the value of the Q parameter that is given by [51-

54]

2

2 dQ

n

(2.1)

where λ is the wavelength of the light used in reconstruction process, n is the

refractive index of the medium after processing, Λ is the grating period, and

d is the thickness of the medium. If Q >2π, the grating is considered as

“thick”, while for Q <2π, the grating is “thin”.

Figure 2.2 Diffraction in a volume grating

14

For gratings formed by two plane waves in a volume hologram, periodic

index modulation induces an array of equidistant planes as depicted in Fig.

2.2. Let us assume that the number of these planes are infinite along

horizontal direction. The planes can be regarded as specular mirrors, which

reflect (scatter) the incident waves with the same angle. The scattered light

consists of linear superposition of all these reflected plane waves. The

diffracted beams are found when all these reflected plane waves add up

constructively under the Bragg’s law which is given by [51-54]

2 sin .d n (2.2)

In a thin grating, the grating dimension in the transverse dimension is

relatively small, compared with the wavelength of light, thus it acts as slits

diffraction rather scattering. Thus, there are a series of diffraction orders for

any given angle of incidence. On the other hand, the thick grating only has a

single diffraction order. This is the main difference between a thin grating

and a thick grating.

2.2.1 Diffraction efficiency: coupled-wave theory

The high diffraction efficiency of the volume hologram makes it practical to

be used as various optical elements, which are called HOE. Two main factors

for HOE design are diffraction efficiency and diffraction direction of the

reconstruction beam. The former one can be analyzed by coupled-wave

theory and the latter one can be analyzed by vector space diagram.

Coupled-wave theory is a useful theoretical basis to design the HOE

15

[51-54]. The Coupled-wave theory mainly consists of three steps. The first

one is modeling of hologram gratings such as axes setting. In this

dissertation, the z-axis is chosen perpendicular to the surface of the medium,

the x-axis in the plane of incidence and parallel to the medium boundaries

and the y-axis perpendicular to the paper as shown in Fig. 2.3.

Figure 2.3 Model of diffraction in a volume grating.

The next step is deriving the coupled equations, which describes

amplitude exchange between reference wave and diffracted wave by solving

the Helmholtz equation of Eq. (2.3). Details of the derivations are beyond the

scope of the thesis, so I only discuss the most important properties that

16

involve the diffraction efficiency of the volume gratings [51].

2 2 0,E k E (2.3)

where E(x, z) is the complex amplitude of the y-component of the electric

field, which is assumed to be independent of y and to oscillate with angular

frequency ω. The wave number k is spatially modulated by reflective index

and absorption constant which is given by

0 1

0 1

cos

cos ,

n n n K r

K r

(2.4)

where K is grating vector. The hologram is assumed to lie with its face

parallel to the (x, y) plane and to be of thickness d in the z-axis. A number of

assumptions are needed for simplification of solving the wave equation. First,

it is assumed that the hologram is thick enough that the total wave field is

expressed by summation of two complex amplitudes.

exp expU r R z j r S z j r . (2.5)

The first term is gradually depleted by diffraction and absorption, the

other is the first-order Bragg-matched grating order. The wave vector is

that of the probe wave in the absence of coupling, and that the wave vector

of the diffracted wave is given by

K (2.6)

The other assumptions are that absorption in a distance of one

wavelength is very small and that the variations of the refractive index are

17

small compared to its mean. It is now possible to expand and simplify k2 for

use in the wave equation as follows:

2

2

0 0 1 0 1

2

0

cos cos

2 4 cos ,

k k n n K r j a a K r

j a K r

(2.7)

where, β=k0n0 is the propagation constant, and κ is the coupling constant,

given by

0 1 1

1.

2k n ja (2.8)

By substituting the assumed solution Eq. (2.5) and the Eq. (2.7) into the

wave equation Eq. of (2.3), we obtain the Coupled-wave equations, which

are given by

0

0 ,

R

s

c R a R j S

c S a S j R

(2.9)

where ς is called the “detuning parameter,” measuring the extend of Bragg

mismatch of the probe wave, which is given by

22

.2

(2.10)

The abbreviations cR and cS are given by

cos

cos cos ,

R

s

c

Kc

(2.11)

18

where θ and φ are defined in Fig. 2.3.

Diffraction efficiency of transmission grating

In the analysis for both transmission and reflection gratings, the

absorption coefficients are set to α0=α1=0. For a transmission geometry, the

boundary conditions to be applied to the differential equations Eq. (2.9) are

R(0)=1 and S(0)=0. The solution for the diffracted wave S at the exit of the

grating is obtained when z=d. The diffraction efficiency η, which is the

energy ratio between probe wave and diffracted wave, is given by [51-54]

2 2 2 2

2 2 2

sin,

1 /

S d

R d

(2.12)

where

1

R S

n d

c c

, (2.13)

0

2

sin / cos

/ 8 R

Kd

K d nc

(2.14)

Diffraction efficiency of reflection grating

For a reflection grating, the grating planes run nearly parallel with the

face of the recording medium. The boundary conditions are changed to

R(0)=1, and S(d)=0. Solution of the coupled-wave equations for the

reflection grating now yields the following expression for the diffraction

efficiency

19

12

2 2

22 2 2

0 1 (1 / ),

S

R d sh

(2.15)

where

1 ,R S

n d

c c

(2.16)

0

2

sin / 2

/ 8 .

S

S

Kd c

K d nc

(2.17)

These expressions can again be used to formulate rules for the angular

bandwidth and the spectral bandwidth of the grating.

2.2.2 Condition to be met for using holographic optical element in

holographic display

Several simulations are carried out to measure the feasibility of using HOE

in the holographic display. First, the diffraction efficiency distribution

according to different grating formations is analyzed. In the simulation, the

gratings are generated by different angles of signal and reference waves and

each angle varies from 0 to π. Figure 2.4 shows the diffraction efficiency

under Bragg-matched condition for transmission type and reflection type

HOE. As shown in Fig. 2.4, the diffraction efficiency of transmission grating

rapidly fluctuates according to different angles of reference and signal waves

while that of the reflection one shows a good uniformity according to

different angles of recording waves.

20

Figure 2.4 Diffraction efficiency distribution (Bragg-matched condition)

according to the different angles of reference and signal waves. (a)

Transmission grating and (b) reflection grating.

According to the simulation results, the reflection type HOE is more

appropriate to be used in the holographic display for the flowing reasons.

When one fabricates an HOE lens, the signal wave is a spherical wave,

which means the k vectors of the signal wave vary within the range of the

numerical aperture (NA) of the lens. The HOE lens should show the similar

diffraction efficiency for k vectors of the signal wave. As shown in Fig. 2.4,

however, the diffraction efficiency of the transmission grating rapidly varies

depending on the angle of the signal wave. Therefore, transmission type

gratings are not appropriate for HOE lens fabrication in terms of the

diffraction efficiency uniformity. The second reason is the angular selectivity

for the HOE. When a plane wave illuminates an SLM, the pixelated structure

diffracts the plane wave within the diffraction angle of the SLM. It means

that if one uses waves coming from the SLM as the probe wave, some

21

diffracted terms whose wave vector are deviated from the reference wave are

Bragg mismatched to the HOE. If the angular selectivity of the HOE is

sufficient, the deviated waves can be guided to the designed directions with a

uniform diffraction efficiency. Thus, the bandwidth of the angular selectivity:

4κd [53] should be larger than at least the diffraction angle range of the SLM.

Therefore, if one wants to use the HOE for holographic display, it should

satisfy following condition:

1

4 sin .dp

(2.18)

The holographic material: photopolymer used in the experiments of the

dissertation has 16 um width and the SLMs used for holographic display

have 3.74 um and 8 um pixel pitches. As shown in Fig. 2.5, the reflection

type HOE satisfies the condition of Eq. (2.18), while the transmission one

does not.

22

Figure 2.5 Angular selectivity of (a) reflection grating and (b) transmission

grating.

Two additional experiments are carried out to validate the analysis

mentioned above. The angle of reconstructed wave depending on the angular

deviation of the probe wave is given by

1sin sin sin sin ,c p s r (2.19)

where θc is the reconstruction wave’s angle, θp is the angle of probe wave, θs

and θr are the angle of the signal wave and reference wave used in the

recording process, respectively. Figure 2.6 shows the experiment setup. The

direction and the power variation of the reconstruction waves are measured

by changing the probe wave from -10° to +10° about the Bragg-matched

angle. In these experiments, the angle of the reference and signal waves are

45° and 0°, respectively. The angle of probe wave is changed by rotating the

HOE, which is placed on a precise motorized rotation stage. A power-meter

is used to detect the reconstructed wave position and the power. A diode-

23

pumped solid state (DPSS) laser with the wavelength 660 nm was used in the

experiment.

Figure 2.6 Experimental setup to validate the feasibility of HOE for

holographic display.

Figure 2.7 shows the experimental results and corresponding simulation

results. In Fig. 2.7(b), the diffraction efficiency distribution is calculated by

Eq. (2.15) (diffraction efficiency of reflection volume grating).

24

Figure 2.7 Experimental and simulation results to validate the feasibility of

HOE for holographic display. (a) Angular deviation of reconstructed wave

and (b) diffraction efficiency according to the angular deviation of probe

wave.

Let us turn to the point why the HOE is more appropriate to be used in

the holographic display rather than the other conventional display. As shown

in Fig. 2.8, when a point light source is imaged by a normal lens, the angular

resolution of the imaged point is determined by the diameter dL of the lens.

The angular resolution is given by

sin 1.22 .Ld

(2.20)

25

Figure 2.8 Resolving power comparison between a normal lens and an HOE

lens.

From Eq. (2.20), the effective diameter dL in the HOE lens is generally

smaller than that of the normal lens because of the angular selectivity. Even

if we neglect the aberration factor, an HOE lens has worse resolving power

than a normal lens. As shown in Fig. 2.8, only limited range in the HOE lens

performs as a desired lens function with a high diffraction efficiency.

The other reason is the severe aberration of HOE lens. As shown in Fig.

2.9, due to the off-axis HOE recording geometry, the wavefront curvature

26

ratio in the vertical and horizontal directions are different, which means it

suffers from severe astigmatism aberration. In addition, there are some high-

order aberrations caused by the optical elements alignment in the HOE

recording and reconstruction systems. As a result, many kinds of aberrations

occur when the HOE lens is used in the imaging system. The HOE lens is

hardly used in an imaging system because of the difficulty in aberration

compensation. On the other hand, the holographic display is wavefront

modulation based display, so it can conveniently compensate the aberrations

by just modifying the contents. Furthermore, both the HOE and holographic

display use the same light source. Considering all these reasons mentioned

above, holographic display is the ideal system to use HOEs as the optical

element.

Figure 2.9 Aberrations comparison between a normal lens and a HOE lens.

27

2.3 Multifunctional holographic optical element

fabrication and application

2.3.1 Design of mirror-lens holographic optical element

The procedure of the MLHOE recording and reconstruction are illustrated in

Fig. 2.10. Based on the angular multiplexing characteristics of volume

hologram, I record two optical elements in a single holographic recording

material using different recording angles. In the first step, a plane signal

wave (S1) and a plane reference wave (R1) are used to record a mirror

function HOE. In the second step, a spherical signal wave (S2), which is

produced by a lens, and a plane reference wave (R2), are used to record a

lens function HOE at the same recording material [46].

In the reconstruction process, a probe wave, which is identical to the R1,

illuminates the fabricated MLHOE. The HOE diffracts the R1 to the SLM

with the direction of S1. Then, the wave modulated by the SLM is used as

the second probe wave R2′ to illuminate the HOE. Finally, the HOE

performing a lens function converges the beams from SLM into MLHOE’s

focal point, which is indicated as S2′. Note that, except the reasons

mentioned in the previous section, the other reason of using reflection type

HOE is to put the light source at the same side with observers to avoid un-

diffracted laser beam directly coming into users’ eyes.

28

Figure 2.10 Schematic diagram of the MLHOE recording and reconstruction

procedures.

In order to implement a see-through holographic AR display, the optical

axes of SLM and observing direction should be located in off-axis. Therefore,

the waves from the SLM illuminates the MLHOE with an oblique angle. A

measurement is carried out to investigate the aberration caused by tilted

probe wave of the MLHOE, and the experiment result is illustrated in Fig.

2.11.

29

Figure 2.11 Experiment to investigate aberrations of HOE lens in hologram

reconstruction procedure.

The hologram of a test pattern shown in the Fig. 2.11 is reconstructed in

the experiment. In Fourier hologram, the spatial spectral bandwidth is in

inverse proportion to the pixel pitch of the SLM. In normal case, the size of

viewing window of reconstructed hologram should be the same in horizontal

and vertical directions. The size of the viewing window along both the

directions are defined as wx,y=𝜆f/px,y, where 𝜆 is wavelength of light source, f

is focal length of Fourier transforming lens, and px and py are pixel pitches of

SLM in the horizontal and vertical directions, respectively [36]. However,

the viewing window is not a square shape when I use the MLHOE to

reconstruct the Fourier hologram as shown in Fig. 2.11. From the experiment,

the ratio of the viewing window size in horizontal and vertical directions is

measured as wx:wy=cos𝜃:1, where 𝜃 (𝜃<90°) is the angle of R2′. It means that

the wavefront from the SLM indeed have aberrations such as astigmatism

after being diffracted by the MLHOE. In order to reconstruct the desired

wave field, I need to conduct a wavefront compensation method to the

hologram, which is introduced in the section 2.3.3.

30

2.3.2 Fabrication of mirror-lens holographic optical element

There are various kinds of holographic materials for implementing the HOE:

the silver halide, photorefractive polymer, photopolymer, and so on. [55-60].

Among them, since HOEs implemented on the photopolymer have a thin,

transparent and high diffraction efficiency, many studies have been reported

using photopolymer based HOE to be transparent image combiners for AR

devices. [59-64].

In the case of recording two holograms in a single HOE, the exposure

energy of each one should be appropriately adjusted to achieve the similar

diffraction efficiency. For this reason, diffraction efficiency distribution

according to gradually increased exposure energy is measured as shown in

Fig. 2.12. During the HOE recording process, interference patterns between

signal and reference waves are stored in the photopolymer by modulating its

transmittance or refractive index. Exposed light on the interference pattern

activates the monomers to polymers. Finally, the difference of the refractive

index between the polymerized area and the original material leads to the

modulation of refractive index. Therefore, if the same amount of monomers

can react for recording each optical element, it is possible to make balanced

diffraction efficiencies between them. The optimized exposure energies for

the two steps are determined based on the measured data shown in Figure

2.12.

31

Figure 2.12 Diffraction efficiency distribution according to exposure

energies.

As shown in the result, the diffraction efficiency rapidly grows as the

exposure energy increases at the beginning, and then the diffraction

efficiency gradually reaches saturation. The diffraction efficiency (η) can be

calculated by Eq. (2.21)

,D

T D

I

I I

(2.21)

where ID and IT represent the power of diffracted and transmitted waves,

respectively. The energy absorption and reflection on the photopolymer and

substrate are neglected in this analysis. As shown in Fig. 2.12, in order to get

50% diffraction efficiency for each one, the exposure energy of the first step

(marked in green) is set to 40mJ/cm2 and that of the second one (marked in

red) is over 182.4mJ/cm2, which is the saturation energy level of the

photopolymer.

The other issue in recording multiple optical elements in a single HOE

by angular multiplexing technique is that the angle of two probe waves

should be sufficiently separated to prevent crosstalk of two optical elements.

32

The probe waves used in the two steps are set to ±45° about the normal of

the photopolymer. Figures 2.13(a) and (b) illustrate the experimental setup

for recording the mirror HOE and the lens HOE, respectively. Since the

recording conditions in the two steps are different, mirror 1(M1) and mirror 3

(M3) are set to be adjustable. A precise rotation stage on a linear stage is used

to handle the position and the angle of M1, which provides the signal waves

for both steps; a rotation stage is used to adjust the angle of M3, which

provides the reference waves of both steps. The principle angles of S1 and

S2 are 45° and 0° to the normal of photopolymer, respectively. The focal

length of the lens that provides spherical signal wave is 300 mm. A plano-

convex lens is used in the second recording step of the MLHOE. It is better

to make the plane wave go into the convex side of the lens and spherical

wave come out from the flat side, because the aberration of the doublet lens

is only compensated along a single direction. In addition, since the S2′ is a

converging spherical wave, the region of the recorded gratings increases by

putting the HOE closer to the lens. Therefore, the photopolymer is placed

very close to the flat side of the plano-convex lens in the actual practice.

33

Figure 2.13 Experiment setup for (a) the 1st step and (b) the 2nd step of the

MLHOE recording process.

2.3.3 System implementation and experimental results

Figure 2.14 shows the photograph of the prototype holographic AR display

system. R1 satisfying the Bragg condition of the mirror HOE is the light

source, and then S1 is diffracted to the reflection type SLM. As shown in Fig.

2.14, the real-world scene, which does not satisfy Bragg-matched condition,

can be clearly observed through the transparent MLHOE.

34

Figure 2.14 Prototype of the compact holographic display using MLHOE.

A DPSS laser with the wavelength 660 nm is used as the light source for

all the experiments. The SLM, which is disassembled from a Sony VPL-

VM300ES projector, is used as the SLM in the holographic display. The

number of the pixels is 4096×2160 and the pixel pitch is 3.74 𝜇m. In general,

SLMs used in projectors are manufactured for amplitude modulation, thus it

is better to use an amplitude type CGH encoding scheme in the experiments

[65-67]. Two polarizers, which are orthogonal to each other, are placed in

front and rear sides of the SLM, so that the SLM can work in amplitude

mode. A half-band filter is positioned in the Fourier plane, so that the DC and

conjugate terms can be eliminated. Figure 2.15 shows the resultant

holograms with the real-world scene captured using the proposed

holographic AR display.

35

Figure 2.15 Experimental results: camera focus at (a) the Fourier plane of the

SLM, (b) the MLHOE, and (c) the real object “Rubik’s cube,” (d) Camera is

set to have a large depth of focus.

The CGH of the polygon object “Cube”, which consists of triangular

meshes, is calculated by using the algorithm proposed by Matsushima et al.

[68-70]. Depending on the relative position of each triangle aperture with

respect to hologram plane, all the triangle meshes’ angular spectrum are

calculated and mapped in the Fourier domain by 3D coordinates rotation and

shift. In the reconstructed images, the distance between center of “CUBE”

and the letter “SNU” is 3 mm, and “SNU” is reconstructed at the Fourier

plane in which the half band filter (a business card) is located. The

36

accommodation effect is demonstrated at four different foci. In Figs. 2.15

(a)-(c), the camera is focused at the Fourier plane, MLHOE and the real-

word object “Rubik’s Cube”, respectively. Figure 2.15(d) is captured with an

enlarged DOF, so that the reconstructed holograms and the “Rubik’s Cube”

can be observed in one scene. The experimental results verify the MLHOE

can function as multiple optical elements to reduce the form factor of the

conventional holographic display system. Furthermore, the transparent

characteristics of the photopolymer enables us to get a see-through

observation condition.

37

2.4 Wavefront aberration compensation method for

holographic optical element

The Zernike polynomial functions are convenient way to characterize

intuitive optical defects such as defocus, astigmatism, coma and distortion

etc. [71,72]. An HOE lens for imaging system generally has severe

aberrations because of the recording condition. Thus, aberration

compensation is an essential process if one wants to use HOE as the imaging

optical elements. In holographic display, since the SLM modulates the

wavefront to reconstruct desired optical field, it is convenient to compensate

wavefront aberrations. Prerequisites for carrying out the pre-compensation is

acquisition of the wavefront information of the HOE. Digital holographic

capturing technic is a well-known method to optical field acquisition method.

There are generally two different optical configurations for the digital

hologram capturing: off-axis and in-axis configurations. The off-axis

geometry is an effective configuration for separating the zeroth-order light

and the twin image [74-76]. However, it is difficult to record a high-quality

off-axis Fresnel hologram because of the limitation of the narrow bandwidth

of the CCD. Thus, the resolution of the reconstructed optical field is not high.

On the other hand, if the on-axis geometry is employed, it is necessary to

eliminate the DC term and the twin noise [74]. Among the various available

techniques, phase-shifting holography is the most widely used technique to

achieve the optical field of the target wave. To get a sufficient good result, a

three or four steps phase shifting technique should be used.

In this work, I use the on-axis digital holography technique based on

38

three-step phase shifting terferometry (PSI) to acquire the wavefront of the

HOE. This interferometer system records three interferograms with

modulating phase of reference wave. In doing so, irradiance of Newton’s

ring-shaped interference pattern is captured by a CCD camera. From three

captured interference patters, the phase information can be recovered through

a simple calculation. The calculated value of phase is only represented within

2π period due to the periodic property of sinusoidal function. The phase

delay of lens actually exceeds the range 2π and is continuous. Therefore, the

physically continuous phase variation of lens is required to be reconstruct by

the unwrapping processing. The unwrapped phase can be expressed using

Zernike polynomials that are efficiently used to represent optical wavefront

aberration of circle-shaped lens. The correction wavefront is constructed by

conjugating to aberration term of the Zernike polynomials. The phase used

for the compensation is modulated by an SLM, which displays images with

gray-scale value corresponding to the phase of correction wavefront before

passing through the Lens HOE (LHOE). As a result, the modulated

wavefront contains only optical property of lens and aberrations of the

LHOE are eliminated after passing through the LHOE. It is possible to form

a sharp focus spot which is similar to a conventional lens. We expect that the

proposed method can mitigate the degradation of image quality by wavefront

aberration.

To check the feasibility of the proposed method, I implemented the

experimental setup, which consists of a 532 nm green laser source, an LHOE,

a phase SLM, a beam splitter, mirror mounted on a piezo actuator and a CCD

camera as shown in Fig. 2.16. I used a 532 nm green laser for the coherent

39

light source. The effective size of CCD is 10 mm (H) 8 mm (V), the

number of pixels is 2448 (H) 2048 (V) and pixel pitch is 3.45 um. The

SLM is an LCoS (liquid crystal on silicon) reflection type that can modulate

the phase of input beam from 0 to 2π, which has a size of 15.3 mm (H)

8.98 mm (V) and a resolution of 4094 (H) 2400 (V).

Figure 2.16 Experimental setup.

The interferometer for the wavefront detecting experiment took the

form of a Mach-Zehnder interferometer. To control the phase of reference

wave, a piezo-electric actuator is attached to the mirror in the optical path

through which the reference beam passed. A piezo actuator is the optical

element that converts a high voltage signal into a physical displacement in

sub-wavelength units. The movement of mirror is precisely controlled to

correspond to a specific phase value for solving the three-step phase shifting

algorithm. For example, the phase differences (φ) between the reference and

the signal waves are -π/3, π, +π/3 that are mainly used in three-step PSI. The

40

optical path difference (OPD) is a value corresponding to 88 nm mirror

movement back and forth from any reference point. The interference fringe

pattern of the reference and the signal wave is captured by the CCD camera

in the form of Newton’s ring, and the brightness of the concentric ring

changes according to the phase difference of the two beams. The CCD is

placed with on-axis line to the interference pattern. The wavefront phase of

the LHOE at the CCD detection position is extracted by the following

formula by using irradiance (In) of three interference patterns captured by

CCD.

sig 0 1 2

2π 4πW = I +I exp i +I exp i /3

3 3,

(2.22)

The original wavefront of LHOE can be found by numerical back-

propagation of the calculated wavefront at the CCD location based on

angular spectrum method. The wavefront propagation is simulated using

MATLAB software to find the wavefront at position of the LHOE. To

recover the physical phase of LHOE wavefront, the 2D Goldstein phase

unwrapping algorithm is applied to the removal of inherent 2π ambiguity.

Figure 2.17(a) and (b) show wrapped phase and the Zernike polynomials

extracted from the unwrapped phase. The vertical astigmatism is as shown in

Fig. 2.17(c).

41

Figure 2.17 (a) Extracted phase of LHOE (b) Zernike polynomials (c)

Zernike polynomials of aberration term.

Figure 2.18 shows the Zernike polynomials of correction wavefront that

compensates the wavefront aberration of the LHOE. In order to modulate by

using a SLM, the phase of the correction wavefront is required to convert to

an 8-bit grayscale image as shown in Fig. 2.18(c).

Figure 2.18 Zernike polynomials of correction wavefront (b) phase of

wavefront (c) converted gray-scale image.

Figure 2.19 shows the captured image of point spread function (PSF) at

the focal distance. Through these results, we confirm that the correction

wavefront has a smaller and sharper PSF than the uncompensated wavefront.

42

Through the experiments, the feasibility of the proposed method is verified

and it is numerically confirmed that the aberration of LHOE on Bragg

mismatching condition is relatively higher than that on Bragg matching by

using a Zernike polynomials, as shown in the Table 2.1.

Figure 2.19 PSF images (a) before compensation on Bragg matching (b) after

compensation on Bragg matching (c) before compensation on Bragg

mismatching (d) after compensation on Bragg mismatching.

Table 2.1 Zernike polynomial coefficients of compensation

Noll index (j) Classical name Bragg matching

(θi = 45°)

Bragg matching

(θi = 46°)

Z5 Oblique

astigmatism 0 -2

Z6 Vertical

astigmatism -10 -12

Z7 Vertical coma 0.3 -0.7

Z8 Horizontal coma 0.2 0.5

43

The proposed method is possible to compensate aberration caused by

physical deformation such as shrinkage or shear of the HOE after recording

process. In the experiments, the wavefront of off-axis HOE is precisely

measured by using PSI. The result of wavefront propagation simulation using

experimental data shows consistent results of actual case. Lasty, it shows that

wavefront aberrations are efficiently removed by applying the correction

wavefront.

44

2.5 Summary and discussion

In summary, an image combiner, which is realized by a multifunctional HOE

is proposed to reduce the form factor of the holographic display. The single

image combiner called MLHOE performs the optical functions of a mirror

and a lens at the same time. Before fabricating the proposed MLHOE, I

analyzed the feasibility of using HOE for holographic display based on the

coupled-wave theory. The optical characteristics of the hologram recording

material (photopolymer), supports the validity of the MLHOE for a see-

through near-eye holography.

In addition, the wavefront aberration of the system can be conveniently

compensated by only modifying the CGH, since holography is basically a

display that controls the wavefront of light. To extract the wavefront

information of the HOE lens, I use a digital holography technique to capture

the fringe pattern generated from the HOE; and the obtained Zerinike

coefficient enable us to compensate the CGH.

Furthermore, it could be possible to implement a full-color holographic

AR display system by using a high speed framerate SLM and a color HOE

which is recorded by three (red, green, blue) lasers. I expect the proposed

method can be a solution for developing a compact see-through holographic

AR displays such as holographic HMD and head up displays.

45

Chapter 3 Field of view expansion of

near-eye holographic display

3.1 Overview of field of view expansion for near-eye

holography

3.1.1 Investigation on constrains in expanding field of view for

near-eye holography

In holographic display, diffraction angle of an SLM determines the FOV,

which is defined by

1sinp

(3.1)

where λ is the wavelength of light source, and p is the pixel pitch of the SLM.

So far, the pixel pitch of commonly used SLMs for holographic display is

around 3.5 um - 12 um [39-47]. Thus, the FOV of the conventional

holographic display is generally less than 10°. As shown in Fig. 3.1, the

optical designs used for near-eye holography can be divided into two types:

pupil relay and beam converging.

46

Figure 3.1 Optical designs for near-eye holography.

In the pupil relay method, a 4-f system is used to relay the de-magnified

SLM to the pupil of the observer. To get a wider FOV, the SLM is generally

demagnified based on the magnification between the two lenses used for the

4-f system. In this case, the diffraction angle θ of the SLM determines the

FOV; and SLM size determines the eyebox. It means FOV, which is

determined by bandwidth of the display is inverse proportional to the pixel

pitch p; and eyebox size, which is determined by spatial size of the display is

proportional to p.

On the other hand, in the second case, all the relations are reversed. In

this case, the FOV is determined by the space size, which is proportional to

p; and eyebox size is determined by the bandwidth, which is inversely

proportional to p. The viewing characteristics of the two kinds of near-eye

holography are listed in Table 3.1, where p′ shown in the table is pixel pitch

47

of the demagnified SLM.

Table 3.1 Definition of the FOV and eyebox size in the two types of near-eye

holography.

FOV Eyebox size

Pupil relay sin-1(λ/p′) p′×n

Beam convergence tan-1(p×n/f) λf/(2×p)

As shown in Table 3.1, the FOV and eyebox size always have tradeoff

relation. Recent trend for expanding the field of view is by sacrificing the

eyebox size i.e. to reduce the effective pixel pitch of the SLM [50]. The

schematic diagram and the experiment setup are illustrated in Fig. 3.2.

Figure 3.2 Experiment schematic diagram and setup of a wide FOV near-

holographic display.

48

An SLM with 8 um pixel pitch and 1920×1080 pixel number, which is

disassembled from an Epson projector, is used to display the CGH. A set of

polarizers are placed perpendicular to each other so that the SLM can

function as the amplitude mode. A lens placed before the SLM is used to

produce a spherical reference wave to the SLM. A high power lens with 50

mm focal length is used as the eyepiece lens. The configuration of the system

enables us to get a 50° FOV and 1.2 mm eyebox.

Figure 3.3 Experimental results: (a) reconstructed image when the camera

focuses at 5 diopter. (b) Reconstructed image when the camera focuses at 0

diopter.

Figure 3.3 shows the captured images through the experiment. The

result shows us that even if the camera focus changes from 5 diopter to the

infinity, the reconstructed image still maintains the same resolution, which is

49

called all-in focus. If the eyebox size is reduced to a certain criterion, the

DOF becomes very long, which is the same as the Maxwellian view. As a

result, the display loses the accommodation effect. Therefore, the eyebox

size should not be reduced arbitrarily for FOV expansion. Figure 3.4 shows

the variation of FOV and DOF depending on different size of the eyebox. In

this analysis, the pixel pitch of the SLM is specified as 8 um.

Figure 3.4 FOV and DOF variation according to eyebox size in holographic

display. (SLM pixel number: 1920×1080, pixel pitch 8 um)

In the region marked in green, the pixel pitch of the SLM is reduced to

be a subwavelength scale. Thus, we cannot use the conventional diffraction

theory in this region. In the region marked in yellow, the eyebox size

50

satisfies the all-in focus condition so it loses the accommodation effect [77-

80]. In summary, the maximum FOV that we can get by reducing the eyebox

size is about 37° (SLM: 1920×1080 pixel number & 8 um pixel size). The

FOV cannot be expanded without losing the size of eyebox if the pixel

number i.e. SBP, is fixed. For an AR display system, the FOV is not

supposed to be very large, because the real scene already occupies the

majority FOV. Let us assume the necessary FOV is 30°. In this case, a

reasonable eyebox size should be 6 mm so that it can support eye rotation in

the FOV range. In this case, if we use an SLM having 1920×1080 pixel

number and 8 um pixel pitch, the SBP should be enlarged 3 times.

Therefore, to solve the fundamental problem of FOV issue, the SBP

must be enhanced. In the next section, I introduce the previous SBP

enhancement methods and two novel proposed methods using high-order

diffraction of an SLM.

3.1.2 Previous space bandwidth product enhancement methods for

holographic display

SBP is an appropriate metric to evaluate the quality of holographic display.

The SBP of a holographic display is defined as the product of the physical

dimension and the corresponding 2D bandwidth of the display.

Several studies were carried out for obtaining a high SBP to alleviate

the issues about FOV and viewing window in the conventional holographic

display i.e. “far-eye holography” [36-42]. A common way to achieve the high

SBP is using plural SLMs in a system. A typical method that tiles multiple

51

SLMs without gap between the adjacent ones, is arranging them at both sides

of a sufficient large beam splitter with regular intervals [36, 37]. Fukaya et al.

demonstrated a viewing angle enlarged holographic display by tiling three

SLMs in Fourier hologram configuration. Later, the system was improved by

tiling five SLMs with more pixel counts. Kozaki et al. proposed a wide

viewing angle holographic display system using numerous SLMs in a curved

structure [38], which does not need a large size Fourier transforming optics

to cover all the SLMs. Han et al. also proposed a holographic stereogram

system using curved array of a number of SLMs [39]. In this method, in

order to remove the gaps resulted from pupils of the transfer lenses, three

sub-regions of an SLM were rearranged to be positioned in contact with each

other diagonally. Actually, the SBP of a single SLM was not increased

because the pixel count in the vertical direction was sacrificed. Other

methods for enlarging the viewing angle of the holographic display, which

focused on naked eye observation, were proposed [40, 41]. Yaras et al.

implemented one of the systems by tiling 9 SLMs; Kozaki et al.

demonstrated another one by using additional SLMs placed in a specified

image space. When a large size object is reconstructed in a small size

viewing window, only a certain part of the object can be observed at a view

point. Furthermore, several methods that arrange multiple 4K×2K-pixels

SLMs in a planar structure to increase the size of viewing window were

reported [43, 44]. For all the methods mentioned above, the inevitable

problem is need of a large number of SLMs. In fact, considering cost,

volume and alignment of the system, using a lot of SLMs to achieve a high

level SBP is difficult to be an ultimate solution. Thus, how to increase the

52

SBP for a single SLM is one of the most important issues in near-eye

holographic display.

53

3.2 Temporal multiplexing of high-order diffraction

guided by holographic attenuating mirror

In this section, the SBP improvement of a single SLM in the holographic

display is proposed by using multiple diffraction orders. The principle of the

proposed method is illustrated in Fig. 3.5. The adopted diffraction orders are

arranged with the same intensity and propagation direction using guiding

optics and temporal-multiplexing system, so that a several-fold increase in

SBP of a single SLM is achieved. The guiding optics system consists of

HOE and a set of relay lenses. The HOE, which functions as both attenuator

and mirror, enables us to adjust both the intensity and propagation direction

of the reconstructed beams. Thus, the guiding optics system can be

simplified by utilizing the features of HOE. Meanwhile, the relay lens

system is used for obtaining a seamless arrangement of the guided terms.

Figure 3.5 Schematic diagram of proposed method.

54

3.2.1 Design of the holographic attenuating mirror

In the actual practices, HOE recording material has a grating detuning effect,

which comes from the dimensional shrinkage induced by the chemical

reactions during the holographic recording procedure [81, 82]. As a result,

the Bragg condition for volume hologram is mismatched and the recorded

information cannot be readout completely. Since the proposed method

requires a uniform intensity and a uniform diffraction direction for the

different diffraction orders, the grating detuning effect is an essential factor

to be considered. The grating detuning effect caused by shrinkage of the

material is illustrated as Fig. 3.6. In this chapter, the v and u stand for the

thickness and transverse directions of the holographic material, respectively.

Figure 3.6 Grating detuning effect caused by dimensional shrinkage.

This work also adopts reflection type holograms. Wave vectors ks, kr,

and kg represent the signal, reference, grating vector; their corresponding

55

vectors after the shrinkage effect are denoted as ks′, kr′, and kg′, respectively,

and their relation in vector space diagram is shown as Fig. 3.7.

Figure 3.7 Vector space diagram for original grating and detuned grating

cases.

According to Bragg’s condition, the recorded grating vector kg is given

by

, g s rk k k (3.1)

where ks and kr are the wave vectors of signal and reference waves inside the

medium.

The outside signal and reference waves are incident to the medium with

the angles 𝜃s and 𝜃r, respectively. 𝜃s and 𝜃r are the angles of those waves

56

after being refracted in the medium, which can be calculated by Snell’s law:

𝜃r·in=sin-1[sin(𝜃r)/ρ] and 𝜃s·in=sin-1[sin(𝜃s)/ρ]. n is the refractive index of the

recording material. Since shrinkage effect occurs after HOE curing process,

the recorded grating vector kg is changed to k′g, which is given by Eq. (3.2).

g

1 1ˆ ˆsin sin cos cos ,

1 1s r s r

u v

u v

s r s sk k k k k

(3.2)

where 𝛼u and 𝛼v are the shrinkage coefficients of the material along u and v

axes, respectively. Due to the shrinkage effect, detuned grating vector k′g

induces phase mismatching when a probe wave, which is identical to the

reference wave in the recording process, is illuminated to the HOE. Due to

the grating detuning effect, the angles of probe and reconstruction waves to

generate maximum diffraction efficiency are deviated from those of original

ones. In order to get the Bragg-matching condition, the amount of the phase

mismatching should be zero. In the actual display process, the directions of

probe wave and reconstructed wave have been specified, which means the

grating formation has been determined. Thus, the pre-compensated reference

and signal waves can be determined experimentally. The recorded gratings

should be detuned to match the desired ones after the curing process. There

are several ways to compensate the grating vector such as angular deviation

method and wavelength deviation method. We can flexibly use any one in

the specific case if the formation of the desired gratings is specified.

As shown in Fig. 3.8, HOE1, HOE2, and HOE3 are designed to align -1st,

0th, and +1st diffraction orders of the SLM, respectively. Mirror 1 (M1) and

mirror 2 (M2) enable us to produce reference and signal waves by

57

appropriately adjusting the positions and angles. The reference waves with

the angles 𝜃r1, 𝜃r2, 𝜃r3, and signal waves with the angles 𝜃s1, 𝜃s2, 𝜃s3 are used

for recording HOE1, HOE2, HOE3, respectively. One of the key advantages

of HOE is the adjustable diffraction efficiency, which follows the dosage

used in the recording process. In order to keep the uniform intensity of the

guided order terms, the exposure energy for recording each HOE should be

handled according to the energy ratio between different diffraction orders of

the SLM.

Figure 3.8 Schematic of the (a) recording and (b) reconstruction processes.

To control the diffraction efficiency ratio correctly, the energy ratio

between the different diffraction orders of an SLM should be investigated.

The SLM with periodic pixelated structure can be regarded as a set of square

gratings. In a single period, the formula for one dimensional amplitude

transmittance function of the SLM is given by Eq. (3.3)

58

, / 2

0, / 2 / 2,

C x at x

a x p

(3.3)

where a is size of the pixel aperture, p is pixel pitch of the SLM, and C is a

constant. The square grating can be regarded as superposition of a series of

sinusoidal gratings with different spatial frequencies. Let us denote f1=1/p as

the fundamental frequency which provides the basic shape of the square

grating. Then, coefficient of the Fourier series, which determines energy of

the diffraction orders, is given by

/2

/2

1exp 2

sin,

a

n na

n

n

T A j f x dxp

A a f a

f a

(3.4)

where fn denotes the frequency of n-th wave, which is an integer multiple of

the fundamental frequency. When a plane wave is incident to a single

sinusoidal amplitude grating having the spatial frequency fn, two diffraction

orders with angles ±𝛽n=sin-1(±𝜆fn) are generated, respectively. It means that

high-order diffractions of an SLM are the terms of the Fourier series carrying

different spatial frequencies. The intensities of the high-order components

are also determined by the spatial frequencies and coefficients of the Fourier

series expansions. Besides the diffraction on an individual pixel, the phase

difference, which is induced by the interference among N pixels should also

be considered. According to Eq. (3.4), the intensity distribution of the high-

order diffraction beams can be calculated by

59

2 2

sin sin / sin sin /.

sin / sin sin /

n n

n

n n

a N dI

a N d

(3.5)

In this work, the dosage for recording the HOE1 and the HOE3 reaches

saturation level, and less dosage, which depends on the energy ratio between

the adopted order terms, is used for recording the HOE2.

In the reconstruction process, the direction of the reconstructed waves

are specified to the same direction. In this case, the size of the reconstructed

beam is determined by the angles of the probe wave and reconstructed wave

of the HOE. In order to obtain a uniform size of the three diffraction order

images, three HOEs are positioned according to the propagation directions of

the three-order terms. If 𝜃r2 is given, tilting angles of the HOE1 and the HOE3

can be calculated as

2 1 21

1

2

2 2 11

3

2

sin costan

cos

cos sintan .

cos

r r

r

r

r

(3.6)

For convenience of the alignment in the display process, HOE2 is set to

be parallel with x axis and 𝜃p2 is specified as 45°. Thus, dx and dz indicated in

Fig. 3.8 (b) are equal to each other. The distance between the centers of the

SLM and the HOE2 is given by d=(dx/cos𝜃p)/tan𝜑, thus the positions of the

HOEs for the reconstruction process can be determined.

The three different diffraction orders of the SLM are located at

difference depth planes because of the path length difference dz. Therefore, a

60

set of relay lenses having different focal lengths are required for imaging

them at the same depth plane with the same size.

3.2.2 System implementation and experiment results

Figure 3.9 shows the dosage curve of the photopolymer. The diffraction

efficiency were saturated when the dosage is over 10 mJ/cm2 in this

photopolymer (different photopolymer to the one used in the previous

Chapter 2). According to Eq. (3.5) and the SLM specifications, the energy

ratio between the ±1st and 0th order diffraction beams is calculated as ~0.37.

As shown in Fig. 3.19, the exposure energies for recording the HOE1 and

HOE3 are marked in red circle; and the blue one is for HOE2. The exposure

energies for recording HOE1 and HOE3 reached at saturation level, so that

they are able to provide a uniform diffraction efficiency.

Figure 3.9 Diffraction efficiency according to exposure energy.

In order to get Bragg-matching condition, the angles of the reference

and signal waves were pre-compensated. In the actual HOE recording

61

experiment, the actually used angles of reference and signal waves were as

follows: 𝜃r1=49.8°, 𝜃r2=50.3°, 𝜃r3=51°, and 𝜃s1=6°, 𝜃s2=3.5°, 𝜃s3=4°; the

tilting angle of the HOE1 and HOE3 were 𝜙1=𝜙3=4.3°.

Figure 3.10 Aligned three diffraction orders by the HOEs.

In the reconstruction process, the intervals of the adjacent HOEs, in x

and z axes were both dx=25.4 mm, thus the distance from the center of SLM

to that of the HOE2 was 461.5 mm, and 𝜃p1, 𝜃p2, and 𝜃p3, were all 45°. The

horizontal size of the SLM was 16.32 mm, then the images size after being

guided by the corresponding HOEs were all w′=23.08 mm. Figure 3.11

shows the guided three diffraction orders of the SLM using the designed

HOEs. The experiment verifies that the HOE is possible to guide them to an

identical direction (i.e. z axis) with a uniform intensity.

An experiment was carried out for investigating the aberration of the

reconstructed hologram. A normal Fresnel zone plate (FZP) propagated to

the HOE by 𝜃p=45°. A screen was placed on an optical rail to adjust the

62

position along z axis. Figure 3.11 shows the experimental configuration,

setup and the captured reconstruction images at different depths.

Figure 3.11 Measurement of FZP aberration. (a) Configuration and (b) the

experimental setup. (c) Captured reconstruction images.

Figure 3.11(c) shows that the reconstructed image is merely focused in

the vertical direction at the depth plane z1, which was the original focal plane

of the FZP. Since, the FZP was stretched in the horizontal direction by use of

the HOE, the focal length along the vertical direction maintained unchanged.

As shown in Fig. 3.11(c), the wavefront was a diverging spherical wave

along the vertical direction and a converging spherical wave along the

horizontal direction within the range from z1 to z3. It means that the focal

length of the FZP in the horizontal direction was extended. The aberration

compensation method mentioned in Chapter 2 is used, so that a target spot

can be focused at the correct depth plane. Figure 3.12 shows the

experimental results for verifying the compensation algorithm.

63

Figure 3.12 Comparison of the reconstructed holograms: (a) without using

the compensation method and (b) using the compensation method.

Figure 3.13 shows the configuration of the proposed holographic

display system. In order to remove the gap between adjacent order terms in

the display process, the relay lenses L1 and L2 were designed as square shape.

The three diffraction order images can be regarded as three virtual SLMs

located at different depths along z axis. Thus, a set of relay lenses (L1), which

have the different focal lengths, were used for imaging them at the same

64

depth plane and with the same magnification. The parameters of the relay

lenses and the virtual SLMs are listed in Table 3.2.

Figure 3.13 Configuration of proposed holographic display system.

Table 3.2 Parameters of the relay lenses and virtual SLMs

Relay lenses Virtual SLMs

Item

Center

coordinates

in x-z plane

(mm)

Width(w)/Diameter(d)

(mm2)

Focal

length

(mm)

Item

Center

coordinates

in x-z plane

(mm)

L1,1 (-25.4,-23.8) 25.4(w) 341 SLM1 (-25.4,-

705.8)

L1,2 (0,-10.4) 25.4(w) 335 SLM2 (0,-680.4)

L1,3 (25.4,0) 25.4(w) 328 SLM3 (25.4,-655)

L2,1 (-25.4,691) 25.4(w) 367

L2,2 (0,691) 25.4(w) 367

L2,1 (25.4,691) 25.4(w) 367

L3 (0,991) 127(d) 300

L4 (0,1491) 50.8(d) 200

In order to increase SBP, they are displayed using a temporal-

multiplexing technic. In the system, three electro-shutters (Thorlabs Co.,

65

SHB05T high speed shutter) whose maximum operation speed was 15 Hz,

were synchronized with the corresponding diffraction order, and an Arduino

UNO R3 micro controller board was used for sending the signal to the

shutters. According to the frequency specified at the output port of the

control board, it sent on/off signals to the shutters and the computer

simultaneously. OpenGL and OpenCV library were used for enhancing the

display speed of the holograms. The size of the eyebox size and FOV the

display were 50.8 mm and 4.3°. The FOV and eyebox size can be adjusted to

optimization values for near-eye holography. As a result, the display system

can provide 30° FOV and 6mm eyebox size. Figure 3.14 shows the benchtop

of the proposed holographic display.

Figure 3.14 Experimental setup of the proposed holographic display system.

In order to confirm the advantage of the SBP enhancement, the

apertures of the three diffraction orders were released sequentially as shown

in Fig. 3.15. By utilizing a temporal-multiplexing technic, holograms of

66

letters “S”, “N”, and “U” were modulated by -1st, 0th, and 1st diffraction

orders of the SLM, respectively. The result shows that it is able to obtain

three-fold increase in SBP of a single SLM.

Figure 3.15 Experimental results of the reconstructed holograms by the

holographic display with increased SBP.

67

3.3 Polarization selective holography

3.3.1 Principle of polarization selective holography

For a liquid crystal (LC) based SLM, the amplitude or phase of input

lights is modulated by sandwiching the SLM between a set of polarizers in

specific states. It means that by adjusting an optical system, which consists

of phase retardation and polarization optical elements, one can get different

responses to the input light. Since the SLM has a pixelated structure, high-

order diffractions, which contain the same information to each other,

inherently occur in the holographic display. Thus, if we set different optical

systems at the path of the different diffraction order paths, a single SLM can

also provide different functions f and g. The configuration of the proposed

method is depicted in Fig. 3.16.

Figure 3.16 Concept of the polarization selective holography.

As shown in Fig. 3.16, polarized lights illuminate an SLM, and a wave

68

plate whose fast axis angle is β0 and wave retardation is γ0 is placed in front

of the SLM. In the paths of the +1st and -1st diffraction order of the SLM,

linear polarizers with the axis direction α1 and α1; wave retarders with the

fast axis angle β1, β2 and phase retardation γ1, γ2 are placed as shown in Fig.

3.16. Thus, the output wave field can be calculated by Jones Matrix [71, 83].

2

out

2

0 0 0

0 0

0 0

0

sin 2cos cos cos 2 sin sin sin 2

2 2 2 2v

sin 2sin sin sin 2 cos cos 2 sin

2 2 2 2

1 1cos cos 2 sin sin sin 2

2 2 2 2 2

1 1sin sin 2 cos cos

2 22 2

i i

i i

i i

i i

e ei i

e ei i

in

0

0

v

2 sin2

(3.7)

Third matrix is the Jones Matrix of an SLM with 45° alignment direction of

LC, where δ is the phase retardation related to voltage applied. Thus, the

amplitude A and phase φ of the output optical field are given by

out

out

cos sin v

cos sin v

A

(3.8)

As a result, if we can find an optimized condition for the 8 parameters

(γ0, β0, γ1, β1, α1, γ2, β2, α2), the optical systems can modulate different optical

waves: A1(x,y) and A2(x,y), according to the corresponding mode. We assume

there is a coordinate system whose horizontal and vertical axes are

represented by A1 and A2, respectively. Thus, the coordinate of a point on the

A1-A2 plane represents that a set of intensity combination for the two

amplitude modulation set. By changing the LC retardation δ from 0 to π (i.e.

69

gray level: 0-255), I can draw an amplitude combination map (ACM) on the

A1-A2 plane. For an 8 bit gray level SLM, the point number of ACM is 256

and the ideal distribution is shown in Fig. 3.17. In this case, two holograms

quantized by 4 bit can be perfectly encoded for the two modes, respectively.

Figure 3.17 The ideal case of ACM for an 8 bit gray level SLM.

If we have two holograms for the modes, the hologram combination

map (HCM) can also be represented on the A1-A2 plane and their coordinate

set on the plane is given by [A1(x,y), A2(x,y)].

The principle to encode the hologram using amplitude modulation is

explained as follows. An optical wave field is generally characterized by two

independent variables at a given point, i.e., the phase and amplitude.

Theoretically, complex SLM can generate perfect optical wave fields with

controlled amplitudes and phases. However, in practices, such a complex

70

SLM has not yet been reported and most available SLMs are designed to be

in the form of amplitude-only or phase-only SLMs with only one degree of

freedom.

In the proposed method, I adopted a single-sideband technic to encode

hologram for an amplitude-only SLM [65-67]. Figure 3.18 shows a 4f optical

system used in the single-sideband technic. g(x,y) represents a desired optical

wave field which is at the image plane and corresponding Fourier transform

is denoted by G(u,v). To separate the signal and conjugate term, G(u,v) is

shifted by carrier frequencies amount ∆u and ∆v along the horizontal and

vertical directions, respectively. Furthermore, the complex amplitude on the

amplitude-only SLM is real value, thus the distribution Gs(u,v) at the Fourier

plane has Hermitian symmetry property as Gs(u,v)= Gs*(-u,-v) and it is given

by

*, , , .sG u v G u u v v G u u v v (3.9)

Figure 3.18 Single-sideband encoding method for an amplitude-only SLM.

Thus, we can get a real-valued function to be encoded on the amplitude-

71

only SLM, which is given by 2Re{F[Gs(u,v)]}. Note that, as the gray level

values represented by the amplitude-only SLM are positive, a bias

component b(x,y) is added to the encoded field to shift to nonnegative.

Summation of bias and un-diffracted DC components generates a delta

function at the center of the Fourier plane. Finally, the reconstructed optical

field is given by

, , exp 2g x y g x y j u u v v . (3.10)

An exponential term that is multiplied to g(x,y), yields bandwidth

shifting which changes propagation direction of the filed in the image plane.

As shown in Fig. 3.16, masking the conjugate and the DC terms limits at

least half of the original bandwidth. The pixel pitch of the SLM is defined as

p, thus the expressible bandwidth is given by 1/2p. As a result, the desired

field g(x,y) propagates with reduced diffraction angle θ≤sin-1(λ/2p).

If the HCM, which is the hologram for amplitude-only SLM, is mapped

to ACM, two different holograms can be generated independently. Thus, how

to find an optimized ACM is the key issue in the proposed method.

3.3.2 Amplitude combination map optimization using genetic

algorithm

We solve the optimization problem using genetic algorithm and it is

explained in this section. Figure 3.19 shows the flow chart of the genetic

algorithm [84, 85]. In the generation stage, the initial population P0 with a

population size n is generated. The population consists of chromosome ci

having 8 parameters c(γ0, β0, γ1, β1, α1, γ2, β2, α2) which are called genes. In

72

the evolution stage, chromosomes having the highest fitness values are

determined by solving their fitness function, which is the most important

factor for determining the performance of the GA.

Figure 3.19 Flow chart of the genetic algorithm.

In the evolution stage, the solution of the optimization problem

converges through a crossover and mutation processes. Several pairs of

chromosomes with highest performance are specified as parents. The

crossover process between the parents is carried out by randomly mixing

their genes i.e. 8 parameters. The reproduced new chromosomes is used to

search the local optima again. Assume that the performance of the first

generation chromosomes are not good enough, it is difficult to get a global

optima just using the crossover process. Thus, a mutation process is essential

in every generation, so that it makes small variations of the present solution

and searches the optimum solution randomly.

73

As a result, optimized variables set is obtained as: β0=11.7°, γ0=165°,

α1=336°, β1=161°, γ1=12°; α2=147°, β2=204°, γ2=58°.

Figure 3.20 ACM and HCM distribution at A1-A2 plane.

As mentioned in the previous section, the distribution of the more complex

or wider ACM is better. The blue line in Fig. 3.20 is the most widely

distributed map. The CGH to be displayed on an SLM is obtained by

mapping the HCM (red dots in Fig. 3.20) to the ACM (green dots in Fig.

3.20). In the mapping process, each point of the HCM finds the closest point

on the ACM, and the searched value of ACM is encoded to the SLM pixel.

3.3.3 Simulation and experimental results

Figure 3.21 shows the simulation result using the proposed method. The

mapped CGH reconstructs different images in accordance with each

modulation mode (+1st diffraction order & -1st diffraction order). The object

used in the simulation and the experiment is a letter “A” and a letter “B”.

74

Figure 3.21 Simulation results using polarization selective holography.

As shown in the result, the left figure has more crosstalk. This is

because the obtained ACM is not symmetric in the horizontal and vertical

axis as shown in Fig. 3.20, which means the ACM can modulate more values

for -1st diffraction order than 1st order diffraction mode.

Figure 3.22 shows the experimental result using the proposed method.

Electronically controllable full-wave liquid crystal variable retarder is used

to precisely adjust the wave retardation and the fast axis angle according to

the optimized variables; and a polarizer with a rotation mount is used to

adjust the polarizer angle.

75

Figure 3.22 (a) Experimental setup. Reconstruction images: (b) -1st

diffraction mode, (c) +1st diffraction mode.

The experimental results also show crosstalk in the two modes. The

reasons are as follows: 1. the performance of the fitness function used in the

genetic algorithm is not good enough to find a perfect optimized solution set.

2. LC retarder is not controlled correctly enough, because I did not have the

lookup table of the retarder in the 532 nm wavelength which is the light

source used in the experiment.

76

3.4 Synthesis of computer generated spherical hologram

Majority of hologram computation methods are introduced based on planar

hologram architecture. Comparing to the planar hologram, a curved

hologram has an important advantage in saving required sampling number.

In this section, I introduce the proposed spherical hologram computation

method of real objects.

3.4.1 Spatial spectral bandwidth of spherical hologram

CGH calculation requires the full 3D information of the objects. For each

object point, the light field distribution at any positions in the space is

defined as below,

exp( )

( , , ) ,i i

i

A jkdH x y z

d

(3.11)

where k=2π/λ is the wave number, λ is the wavelength, Ai is the intensity of

the object point, and di is the length of the vector pointing from the object

point Pi(xi, yi, zi) to the recording point P(x, y, z), which is given by

2 2 2( ) ( ) ( ) .i i i id x x y y z z (3.12)

In order to calculate the light distribution on the spherical surface, it is

advantageous to use the spherical coordinates system, thus the diffraction

field of the spherical hologram is given by

77

1 22

2 2

1 22

2 2

sin cosexp

sin sin cos( , ) ,

sin cos

sin sin cos

i

i

i i

r

i

i i

r xA jk

r y r zH

r x

r y r z

(3.13)

where radius of the hologram is denoted as r, the latitudinal axis is defined as

θ and the azimuthal axis is defined as φ. Figure 3.33 shows one example of

the calculated optical field when a point object is located at (xi, yi, zi)=(0, 0,

0.1m), where λ=532 nm and r=1 m.

Figure 3.23 (a) Position of the object point and the hologram surface. (b)

Amplitude and (c) phase distribution of the spherical hologram.

In order to compute a spherical hologram, it is necessary to analyze the

spatial spectral bandwidth. The sampling theorem should be satisfied in both

78

the latitudinal and azimuthal directions. The local spatial frequency pair (fLθ,

fLφ) of the phase of the spherical hologram Hr(θ, φ) is given by [52]

1( ),

2

1  ( ),

2

L i

L i

f kd

f kd

(3.14)

which leads to

1 22 2 2

1 22 2 2

( cos cos cos sin sin ),

2 sin cos sin sin cos

sin ( sin cos ).

2 sin cos sin sin cos

i i iL

i i i

i iL

i i i

kr x y zf

r x r y r z

r x yf

r x r y r z

(3.15)

Without loss of generality, the object point Pi is assumed to be located

at (xi, yi, zi)=(ro, 0, 0) as denoted in Fig. 3.24. Thus, the local frequencies

corresponding to the point object can be written as

1 22 2

1 22 2

cos cos,

2 2 sin cos

sin sin.

2 2 sin cos

oL

o o

oL

o o

krrf

r rr r

krrf

r rr r

(3.16)

79

Figure 3.24 Object points with the same distance from coordinate origin.

According to Eq. (6), the maximum local spatial frequency │fLθ│max

appears at the position where θ=sin-1(ro/r), φ=0 of the hologram surface, and

is given by

max

.2

oL

krf

(3.17)

In the same way, │fLθ│max is given by

max

.2

oL

krf

(3.18)

For φ direction, the maximum local frequency │fLθ│max

appears at the

position where θ=π/2, φ=sin-1(ro/r) of the hologram surface. The results shown

above indicate that the object point located farther from the coordinate origin

requires higher local frequency for the spherical hologram. The minimum

80

number of hologram points to satisfy the Nyquist sampling theorem in each

axis is given by

max

max

2 ,

4 2 .

L o

L o

N f kr

N f kr

(3.19)

One of the key advantages of the spherical hologram is that the data

amount for a given field of view is smaller than a planar hologram. Suppose

a two-dimensional case where a point object is located at P(xi, yi)= P(d′, 0) to

as shown in Fig. 3.25.

Figure 3.25 Spherical and planar hologram for optical field of the same FOV.

For a 2β FOV, the maximum spatial frequency is given by

│fLθ│max=kd′/2π for a spherical hologram, and │fLy│max=ksinβ/2π for a

planar hologram. Therefore, the number of required samples to represent the

optical field is much larger in planar hologram as depicted in Fig. 3.26.

81

Figure 3.26 Required number of samples for the spherical and the planar

hologram. d′= 0.18 m, r=1 m.

The result reveals that using spherical hologram can significantly

decrease the data amount than using planar hologram in case of large FOV.

3.4.2 Spherical hologram calculation of real objects

In spherical hologram, the 3D information acquisition of the real object is a

key element. In order to extract 360° depth information of a real object, I

captured a real object by placing depth camera in front, rear, left, and right

sides around it. Figure 3.27 shows the four sets of the point cloud, which are

extracted by the depth camera in corresponding directions.

82

Figure 3.27 Extracted depth maps of the object in multiple directions. (a)

Color images of the object, (b) corresponding depth map images.

The obtained point cloud sets need to be brought into a common

coordinate system, since each depth map is captured in its own coordinate

system. In our proposed method, the point cloud registration (PCR)

algorithm is used. The PCR process consists of two steps, i.e. rotating and

shifting as given by

ˆ ˆ ,T oP R P T (3.20)

where ˆoP the initial point cloud, ˆ

TP the registered point cloud, R rotation

matrix, and T is the translation matrix. The rotation matrix R is given by

cos , 0, sin1, 0, 0 cos , sin , 0

0, cos , sin 0, 1, 0 sin , cos , 0

sin , 0, cos 0, 0, 10, sin , cos

y y z z

x x z z

y yx x

R

(3.21)

83

where rotation angles about x, y and z axes are denoted as γx, γy and γz,

respectively. Since the depth camera is set to capture the object around z axis,

γx, and γy are 0° in the proposed method. The point cloud, which is obtained

from front side of the object, is specified as the reference model. Thus, the

rotation angle γz of data obtained from right, rear, and left side of the object

are 90°, 180°, and 270°, respectively. The original point cloud sets and the

rotated ones are shown in Fig. 3.28

Figure 3.28 Rotation of the initial point cloud sets. (a) Initial point clouds

and (b) rotated ones.

The rotated point cloud sets are denoted as 1ˆ

RP , 2ˆ

RP , 3ˆ

RP and 4ˆ

RP ,

respectively. After rotation, the translation vector Ti for each point cloud ˆi RP

is given by

84

1

2 1 2 1 2 1 2

3 1 3 1 3 1 3

4 1 4 1 4 1 4

0,0,0

, ,

, ,

, , ,

x x y y z z

x x y y z z

x x y y z z

T

T m m m m m m

T m m m m m m

T m m m m m m

(3.22)

where, mix, miy and miz denote maximum coordinate values of the rotated

point cloud ˆi RP

in x, y, and z axes. Finally, a 3D model ˆ

TP that contains

360° depth information can be obtained by

1 2 2 3 3 4 4ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ),T R R R RP P P T P T P T

(3.23)

where ∪ stands for union operation. Figure 3.29 shows the 3D model

constructed by PCR.

85

Figure 3.29 Acquired 3D model of the real object.

After registration, visible point set for a given viewpoint needs to be

identified for correct occlusion effect. In our method, the HPR technique is

used [49]. The HPR technique consists of two steps, i.e. inversion flipping

and convex hull extraction. First, the object point cloud ˆTP

is converted by

spherical flipping function, which is expressed as

ˆ

ˆ ˆ ˆ2( ) ,ˆT

f T f T

T

PP P r P

P

(3.24)

86

where 1 2

ˆ ˆ ˆT T TP P P and rf is the longest distance between the

hologram point Hr(θ, φ) and the object point. By using Eq. (3.24), the object

point cloud can be flipped as shown in Fig. 3.30.

Figure 3.30 Spherical flipping of the 3D point cloud by Eq. (3.24). The

object point cloud ˆTP and the flipped point cloud ˆ

fP .

Step 2 is the process of extracting convex hull. Convex hull of the point

set, which is marked in red line in Fig. 3.30, is the smallest convex set that

contains the hologram point and the flipped point cloud. The selected convex

hull points are visible to the given hologram point. Figure 3.31 shows the

result of the HPR. The visible points set of the object at the hologram point

Hr(θ, φ) is denoted as ( , )P̂ .

87

Figure 3.31 (a) Entire points without HPR and (b) the visible point set

processed by HPR at the hologram point Hr(90°, 90°).

In our proposed method, the visible object points are identified for each

hologram point as illustrated in Fig. 3.32.

Figure 3.32 Visible point set for each hologram point.

Thus, a complex value of the hologram point is given by

88

1 22

2 2

1 22

1

2 2

sin cosexp

sin sin cos( , ) .

sin cos

sin sin cos

i

N i i

r i

ii

i i

r xjk

r y r zH A

r x

r y r z

(3.25)

where Ai is intensity of the i-th point of the visible point set ( , )P̂ and the

Cartesian coordinates of the point is (xi, yi, zi). This computation is performed

repeatedly until the optical field values for all spherical hologram points are

determined.

3.4.3 Simulation and experiment results

The proposed method is verified experimentally. First, I show the spherical

holograms for a single object point. Figure 3.33 shows the amplitude and

phase images when the object point is located at several different positions.

The first row images are calculated in the case that the object point is located

at (xi, yi, zi)=(0, 0, 0.01 mm). Second, third and last rows are the amplitude

and phase images when the point located at (xi, yi, zi)=(0, 0.1 mm, 0), (xi, yi,

zi)=(0.1 mm, 0, 0), and (xi, yi, zi)=(0.058 mm, 0.058 mm, 0.058 mm) with the

r=10 mm, respectively. According to Eq. (3.19), the required number of

samples in the spherical hologram is calculated as 1181(θ) ×2362(φ).

89

Figure 3.33 Spherical hologram generated by one point object. (a) Amplitude

and (b) phase images

After confirming the spherical hologram generated by one light point

source, the experiment of spherical hologram of real object is performed. In

the experiment, KINECT sensor is used as the depth camera to extract object

depth information. Recently Microsoft released a non-commercial KINECT

software development kit (SDK) for Window 7. I use C# program language

to run the KINECT SDK. The resolution of each depth map image obtained

using KINECT is 320(V)×240(H). The acquired 3D model of the real object

in Fig. 3.34 is composed of 14531 points while size of the object is limited in

90

the range from -1.4 mm to +1.4 mm in each x, y and z axis. In the

experiment, the radius r of the spherical hologram is set to 95.7 mm, and the

distance between the farthest object point and the coordinate origin is 1.8

mm. Thus, according to Eq. (3.19), the required number of samples in the

spherical hologram is calculated as 21765(θ)×43530(φ). The angular pitch

∆θ and ∆φ are 0.14 mrad, respectively. Figure 3.44 shows amplitude of the

generated spherical hologram. In order to improve the computing efficiency,

the hologram is separated into several parts and five computers are used to

calculate each sub-hologram, and then the completely spherical hologram is

synthesized by combining calculated sub-hologram.

Figure 3.34 Amplitude of the generated spherical hologram.

In order to verify the synthesized hologram, some arbitrary regions of

the spherical hologram are selected and reconstructed. Reconstruction

process of an interest region is shown in Fig. 3.35. The reconstruction plane

ξ-η is set to be normal to the radial line from the coordinate origin to the

91

center of the selected hologram region. The reconstructed optical field U(ξ,η)

is given by

,

1 1 ,

exp( )1( , ) ( , ) cos ,

s wu v

r

u v u v

jkdU H

j d

(3.26)

where s and w are the number of the hologram points in the selected region

in θ and φ axes, Θ is the angle between normal vector of ξ-η plane and vector

pointing from the hologram point P(θ,φ) to reconstruction plane point P′(ξ,η)

is the distance between the two points.

Figure 3.35 Reconstruction process of the selected region.

92

Figure 3.36 shows the reconstruction results of some partial regions of

the spherical hologram, where CR: (θ,φ) is the center region of the selected

curved surface. The first column shows the amplitude, phase and their

reconstruction images of the selected hologram region with CR:

(θ,φ)=(90°,20°), where the hologram size is 230(θ)×230(φ) and size of the

reconstruction plane is 300(θ)×300(φ). Furthermore, I arbitrarily select the

other regions of the hologram with the same size as the first one, for

CR:(110°,110°), CR:(90°,90°), CR:(75°,110°), CR:(70°,270°) and

CR:(90°,270°), respectively. The corresponding amplitude, phase and their

reconstruction images are also shown in Fig. 3.36.

93

Figure 3.36 Reconstruction of the partial regions of the spherical hologram.

(a) Amplitude, (b) phase and (c) reconstruction image of the selected

hologram region.

The result demonstrates that the partial region of the spherical hologram

synthesized by the proposed method can reconstruct corresponding perspective of the

real-existing object successfully. Figure 3.37 shows the effect of HPR process. The

same regions of the proposed spherical hologram and the hologram generated without

using HPR are selected and reconstructed. In Fig. 3.37 (b), because all the object

points are calculated, it is difficult to determine whether the object is looking forwards

94

or backwards at the viewpoint. Furthermore, Fig. 3.37 (a) show that the spherical

hologram generated by the proposed method can clearly portray the shape of the

object, whereas it is difficult to identify which region of the object is reconstructed

when HPR process is at used as shown in Fig. 3.37 (b). Hence, the identification of

visible points is quite an important process for correct occlusion effect in spherical

hologram generation.

Figure 3.37 (a) Reconstruction image from a partial region of the proposed

spherical hologram, (b) reconstruction image from the spherical hologram

generated without using the HPR algorithm.

95

3.5 Summary and discussion

The method that widens FOV by reducing the eyebox size cannot be the

ultimate solution for the FOV issue, because eyebox size should be at least

larger than 1.3 mm to maintain the accommodation effect. Furthermore, the

eyebox size should be determined in accordance with the FOV to support eye

rotation within the entire FOV range. Therefore, increasing the SBP is an

ideal solution. In this chapter, two SPB enhancement methods are introduced.

The first one is temporal multiplexing of high-order diffraction of an SLM.

In this method, three HOEs are designed to guide the different diffraction

orders to an identical direction with the same intensity. A set of relay lenses

and electro-shutters are used to implement the display as if three-SLMs are

arranged with seamless effect. As a result, the demonstrated system achieves

a three-fold increase in SBP of a single SLM.

A main disadvantage of the first method is the requirement of the

mechanical movement in the system. A novel method called polarization

selective holography is proposed. Unlike the conventional SBP enhancement

methods the proposed polarization selective holography does not need to do

temporal multiplexing process which requires mechanical movement. It also

can get two times increase in terms of SBP without using any additional

SLMs. In general, in order to increase the SBP we have to scarify some

parameters. In this method, the scarified parameter is contrast i.e. gray level

of the SLM. If there is an SLM having larger phase retardation such as

2π~4π we can get more complex distribution of the ACM. However, the

more phase variation causes more errors in the amplitude mode holographic

96

display. Therefore, how to decide an SLM’s phase retardation is most

important issue in this method.

A curved surface hologram can provide wider field of view. Thus,

curved holographic display has a great potential to the near-eye holography.

A novel method for generating a spherical hologram of real objects has been

proposed in this section. To my best knowledge, it is the first demonstration

of 360° spherical hologram of real-existing object. A depth camera is used to

extract depth information of the object from different directions. By applying

HPR to the spherical hologram synthesis, the occlusion effect can be

correctly considered in the proposed method. The experimental results

demonstrate that the proposed method is capable of synthesizing a spherical

hologram, which contains 360° field of view information of the real-existing

3D objects successfully. We believe that the proposed method is particularly

useful in building a database for storing the information of real objects’ 360°

optical field, so that we can reconstruct any region of the interest.

97

Chapter 4 Light source selection for

near-eye holographic display

4.1 Coherence of light source

Light source plays a critical role in the holographic displays. Laser is a

common light source used in the holographic display because of its high

temporal and spatial coherence properties. However, the high temporal

coherence induces significant speckles in the reconstructed images, which

affects the image quality greatly. To alleviate the issue, LED that is a

partially coherent light source was attempted as the light source in some

reports. However, the low temporal coherence of the LED could constrain

the reconstructable depth range of the images. In addition, to get a

satisfactory spatial coherency, a spatial filter leads to decrease the

illumination energy. In consequence, the coherence properties of the light

source directly influence the quality of hologram images on both image

sharpness and speckle. This section discusses the fundamental properties of

the coherence of the light source and propose a simple alternative for the

light source issue of near-eye holography.

4.1.1 Temporal coherence

Temporal coherence measures the correlation between waves observed at

different moments in time. When the correlation decreases around 0, the

coherence time tc can be calculated by the phase delay between the two

98

waves. The coherence length Lc is defined as traveled distance of the wave in

tc. According to the temporal coherence, we can know how monochromatic a

light source is. The coherence length of a light source can be obtained based

on the Gaussian emission spectrum, and it is given by [86]

22ln 2cL

n

(4.1)

where n is the refractive index of the medium, λ is the central wavelength

and Δλ is the full width half maximum of the peak in wavelength spectrum.

For measurement of the temporal coherence, we can build a Michelson

interferometer or numerically calculate the Lc by substituting the Δλ, which is

provided by the laser’s manufacturer. Once the fringes are obtained in the

Michelson interferometer, when one of the mirrors is translated along the

beam path gradually, the time for the beam to travel increases and the fringes

become dull and finally are lost, showing temporal coherence.

4.1.2 Spatial coherence

Spatial coherence actually describes how small the size of the light source is.

As shown in Fig. 4.1, light from different parts of the light source propagates

to a screen thus there is a specific correlation between position A and B. Let

us assume that the size of the light source is infinitely small, the optical

waves at position A and B have a constant phase delay, which means they

have a maximum correlation. In this case, the light source is regarded has a

very high spatial coherence.

99

Figure 4.1 Correlation between two light poits from a light source

One way to calculate the degree of spatial coherence is to apply Van

Cittert-Zernike theorem. The theorem deal with that the problem of spatial

coherence by solving the diffraction problem [90]. In this calculation, an

aperture that has the same size with the light source is placed at the position

of the light source. The aperture is then illuminated by a spherical wave,

which is emerged from a point light source located at position A or B. The

optical field on point A of the screen can be found using Fresnel diffraction

formula, which is given by

20

0

; e x p2

e x p

A

A

jkU x x C x

r

jkA x x d

r

(4.2)

where C is a constant, optical field of the aperture is determined by A(ξ), and

xA is the coordinate of point A on the screen. Optical field at the point B

U(x;xB) is calculated as the same manner with Eq. (4.2). Finally, the complex

100

degree of coherence between points A and B is the form of the normalized

diffracted field can be calculated as Eq. (4.2) in [86, 90].

0

0

,

exp

.A B

x

A B

AB

k x xk

r

jkA x x d

r

A d

F A x

A d

(4.2)

101

4.2 Image sharpness for holographic display

In holographic display, the light source illuminates CGH, which is displayed

on an SLM, to reconstruct holographic images. The optical field information

on the SLM diffracts incident light wave to reconstruct the desired

holographic images. For convenience of explanation, a set of double slits

case representing two spatially separated pixels of the SLM, is used for

explaining the reconstructed image quality depending on coherence. Fig.

4.2(a) shows how the temporal coherence influence the interference patterns

sharpness, and Fig. 4.2(b) shows the spatial coherence case. In Fig. 4.2(a), a

light source illuminates the double slits screen with different wavelengths. In

double slit plane, two slits function as two individual light sources

propagating to the second screen. Note that the optical field at the second

plane is calculated by using Fresnel propagation algorithm. The interference

pattern formed by the longer wavelength, which is marked in the red line,

has a longer period. The superposed result shows that the pattern is imposed

by a new envelope curve.

In Fig. 4.2(b), a light source is assumed to have a certain size and two

points on the light source are used in the simulation. Same as the first

simulation, the two light sources also propagate to the double slits screen and

the slits function as the two distinguished light sources to form interference

patterns. Since the two point sources have the same wavelength, the

interference pattern generated by each point source shifted with a constant

phase, thus the superposed interference pattern shows a decreased contrast.

102

Figure 4.2 Comparison of image sharpness using light sources with different

coherencies. (a) How temporal coherence and (b) spatial coherence affect

quality of reconstruction images.

The results reveal that the temporal coherence reshapes it by imposing

an envelope curve while the spatial coherence directly changes the contrast

103

of the interference pattern. If we merely take account the above factors, the

better coherence properties on both temporal and spatial domain can

contribute better reconstruction image qualities. However, another important

factor, i.e. speckle, is also a very important, affecting the holographic image.

Let us consider a wave emitting from a point source and there is a rough

surface on its propagation path. The wave reflected by such a rough surface

is the sum of secondary waves from many independent scattering areas. The

condition that two secondary waves can interfere is that their light path

difference should be shorter than the coherent length, while the coherent

length is directly associated with the temporal coherence.

A simple experiment, which illuminates the same hologram with LD

and an LED, is carried out to compare the reconstruction images. Figure

4.3(a) shows a hologram reconstructed at several different depths by LD and

LED, respectively and Fig. 4.3(b) shows the light bandwidths of the light

sources used in the experiment. Note that, a spatial filter composed of a

micro pinhole and an objective lens is used for the two light sources so that

the experiments are carried out under the same spatial coherence. Since, the

temporal coherence length of the LD is much longer than LED, it can

reconstruct quite a sharp image at a long depth range while the high temporal

coherence leads to the severe speckle noises. On the other hand, the LED

based holographic images show very low speckle noise level, but the

expressible depth range is limited.

104

Figure 4.3 (a) Holograms reconstructed at different depths from an SLM

using LD and LED light source. (b) Spectrum of the light source.

105

4.3 Ideal optical design for LED based near-eye

holography

Table 4.1 Features of the two light sources (same spatial coherence) for

holographic display.

Light source Temporal

coherence

Speckle

noise

Expressible

depth range

Image

sharpness

LD High Server Long High

LED Low Weak Short Depth

Table 4.1 shows the features of the two light sources for holographic display,

when the two light sources are filtered by the same spatial filter. The

inherence properties of light source determine the temporal coherence which

is unable to adjust. It means the speckle noise cannot be removed without

adding optical or mechanical systems. Therefore, the solution proposed in

this section focuses on the adjustable factor, i.e. expressible depth range,

which also determines the image sharpness of LED. Figure 4.4 shows the

schematic diagram of the proposed method. A fiber coupled or a spatial

filtered LED light source is used as the light source to suppress the speckle

noise. As shown in Table 4.1, the expressible depth range maintaining the

sharpness of holographic image is limited in a short range [87-89]. I use a

short focal length, which is shorter than the expressible depth range, and

place the SLM at the focal plane of the eyepiece. In the configuration, the

SLM can reconstruct the hologram from infinity to the nearest visible depth

106

of user by merely reconstructing the image in a short depth range S

according to the equation

1,

near f

S fD D

(4.3)

where f is the focal length of the eyepiece and its dioptric notation is Df . And

Dnear is diopter of the nearest visible distance, which is generally defined as 5

diopter. In the method, the expressible depth range of LED based holography

is defined as 0.17 diopter, thus the power of the eyepiece should be shorter

than 0.17 diopter. To reconstruct the image from infinity to the nearest

visible depth while maintain image sharpness, the reconstruction depth range

S just need to cover from 0 to 14 mm. As shown in Fig. 4.3 (a) LED based

holography is sufficiently able to reconstruct a good quality image in this

range.

Figure 4.4 Optical design for the near-eye holography using a light source

with low temporal coherence.

107

Figure 4.5 shows the experimental results using the proposed method.

The object letters A and B are constructed from 0 to 5 diopter by

reconstructing the hologram from 0 to 14 mm from the SLM. The result

reveals that the proposed LED based holography can have quite a good

image quality for a sufficiently long depth range. The noise-like background

is the DC term, which can be removed using single side band holography

technic. The purpose of the experiment is to show that only using an LED

can also cover the complete depth range for user. As a result, the proposed

method can effectively suppress the speckle noise for near-eye holographic

display.

Figure 4.5 Reconstructed hologram at different depth plane using the

proposed optical configuration.

108

4.4 Summary and discussion

In summary, a light source with high spatial coherence and moderate

temporal coherence can provide a high quality images with good sharpness

and minimum speckle noise in holographic display. To use a low temporal

coherence light source such as LED, a magnification optical system should

be used to mitigate the short expressible depth rage. The experimental results

shows that even if an LED is used as the light source, the holographic

display can also reconstruct the image from infinity to nearest focal plane of

observers. If a light source has the ability that changes the temporal

coherence property, it would be an ideal light source for the near-eye

holographic display.

109

Chapter 5 Conclusion

AR/VR near-eye displays provide a lot of potential to create new life style

for people. To develop an ideal near-eye devices several issues should be

ultimately solved, which are mentioned in the introduction section.

Compared with the other 3D display techniques, holography is the best way

to create natural focus cues. However, the holography technique has not yet

reached a mature step to make consumers widely accept it. This dissertation

has proposed several approaches to solve the fundamental problems of the

near-eye holography. The issues involves form factor reduction, FOV

expansion, and light source selection.

To reduce the form factor of the conventional holographic display, I

used a multifunctional HOE: MLHOE to replace the beam splitter and the

eyepiece lens of the display system. In this study, the optical characteristics

of the HOE are analyzed based on coupled-wave theory. The analysis mainly

targets on the angular selectivity, angular deviation under Bragg-mismatched

condition, and diffraction efficiency uniformity of the HOE. Considering all

of the evidences, I conclude that the HOE can be a very good optical element

for a near-eye holographic display.

For the next issue: FOV, I discussed the topic from the tradeoff between

FOV and eyebox size. The method of reducing the eyebox size to enlarge the

FOV, cannot be the ultimate solution because the accommodation effect

occurs when the eyebox size is larger than a certain criterion. Thus, two SBP

enhancement methods are proposed. The first method uses three different

110

diffraction orders of an SLM to get a three-fold increase in SBP. However,

the requirement of mechanical movement causes the large form factor of the

system. Thus, another SBP enhancement method without any mechanical

movements named polarization selective holography is proposed. In the

method, an SLM generates two different signals with two different

modulation modes. The different modulation modes are realized by using a

wave retarder and a polarizer with different variables. Therefore, by placing

the stationary optical elements at the paths of two different diffraction order

of an SLM, the SLM can indecently modulate the two different holograms.

As a result, I can increase the SBP two times for a single SLM.

Speckle noise issue is also discussed in this dissertation. Use of LED

can suppress the speckle noise in the holographic display. Using the LED,

however, cannot reconstruct images located far from the SLM. Thus, a

magnification optical system should be used to mitigate the short expressible

depth range. If a light source has the ability that can change the temporal

coherence property, it would be an ideal light source for the near-eye

holographic display.

Although the holographic display still has many problems to be solved,

I believe it has chances to succeed in the near-eye display area especially for

AR.

111

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Appendix

Portions of the work discussed in this dissertation are also presented in the

following publications:

[Chapter 2.3] G. Li, D. Lee, Y. Jeong, J. Cho, and B. Lee, “Holographic

display for see-through augmented reality using mirror-lens holographic

optical element,” Optics Letters, vol. 41, no. 26, pp. 2486-2489, 2016

[Chapter 3.2] G. Li, J. Jeong, D. Lee, J. Yeom, C. Jang, S. Lee, and B. Lee,

“Space bandwidth product enhancement of holographic display using high-

order diffraction guided by holographic optical element,” Optics Express, vol.

23, no. 26, pp. 33170-33183, 2015.

[Chapter 5.3] G. Li, A.-H, Phan, N. Kim, and J.-H. Park, “Synthesis of

computer-generated spherical hologram of real object with 360 degree field

of view using a depth camera,” Applied Optics, vol. 52, 3567-3575, 2013.

[Chapter 5.3.3] G. Li, K. Hong, J. Yeom, N. Chen, J.-H. Park, H. Kim,

“Acceleration method for computer generated spherical hologram calculation

of real objects using graphics processing unit,” Chin. Opt. Lett., vol. 12,

article 060016, 2014